3  Differential Geometry

$$ \newcommand{\LetThereBe}[2]{\newcommand{#1}{#2}} \newcommand{\letThereBe}[3]{\newcommand{#1}[#2]{#3}} \newcommand{\ForceToBe}[2]{\renewcommand{#1}{#2}} \newcommand{\forceToBe}[3]{\renewcommand{#1}[#2]{#3}} \newcommand{\MayThereBe}[2]{\newcommand{#1}{#2}} \newcommand{\mayThereBe}[3]{\newcommand{#1}[#2]{#3}} % Declare mathematics (so they can be overwritten for PDF) \newcommand{\declareMathematics}[2]{\DeclareMathOperator{#1}{#2}} \newcommand{\declareMathematicsStar}[2]{\DeclareMathOperator*{#1}{#2}} % striked integral \newcommand{\avint}{\mathop{\mathchoice{\,\rlap{-}\!\!\int} {\rlap{\raise.15em{\scriptstyle -}}\kern-.2em\int} {\rlap{\raise.09em{\scriptscriptstyle -}}\!\int} {\rlap{-}\!\int}}\nolimits} % \d does not work well for PDFs \LetThereBe{\d}{\differential} \LetThereBe{\Im}{\IM} \LetThereBe{\Re}{\RE} \letThereBe{\linefrac}{2}{#1/#2} \LetThereBe{\ExtProd}{\mathsf{\Lambda}} \letThereBe{\unicodeInt}{1}{\mathop{\vcenter{\mathchoice{\huge\unicode{#1}}{\unicode{#1}}{\unicode{#1}}{\unicode{#1}}}}\nolimits} \letThereBe{\Oiint}{1}{\underset{ #1 \;}{ {\rlap{\mspace{1mu} \boldsymbol{\bigcirc}}{\rlap{\int}{\;\int}}} }} \letThereBe{\sOiint}{1}{\unicodeInt{x222F}_{#1}} $$ $$ % Simply for testing \LetThereBe{\foo}{\textrm{FIXME: this is a test!}} % Font styles \letThereBe{\mcal}{1}{\mathcal{#1}} \letThereBe{\chem}{1}{\mathrm{#1}} % Sets \LetThereBe{\C}{\mathbb{C}} \LetThereBe{\R}{\mathbb{R}} \LetThereBe{\Z}{\mathbb{Z}} \LetThereBe{\N}{\mathbb{N}} \LetThereBe{\K}{\mathbb{K}} \LetThereBe{\im}{\mathrm{i}} % Sets from PDEs \LetThereBe{\boundaryOf}{\partial} \letThereBe{\closureOf}{1}{\overline{#1}} \letThereBe{\Contf}{1}{\mcal C^{#1}} \letThereBe{\contf}{2}{\Contf{#2}(#1)} \letThereBe{\compactContf}{2}{\mcal C_c^{#2}(#1)} \letThereBe{\ball}{2}{B\brackets{#1, #2}} \letThereBe{\closedBall}{2}{B\parentheses{#1, #2}} \LetThereBe{\compactEmbed}{\subset\subset} \letThereBe{\inside}{1}{#1^o} \LetThereBe{\neighborhood}{\mcal O} \letThereBe{\neigh}{1}{\neighborhood \brackets{#1}} % Basic notation - vectors and random variables \letThereBe{\vi}{1}{\boldsymbol{#1}} %vector or matrix \letThereBe{\dvi}{1}{\vi{\dot{#1}}} %differentiated vector or matrix \letThereBe{\vii}{1}{\mathbf{#1}} %if \vi doesn't work \letThereBe{\dvii}{1}{\vii{\dot{#1}}} %if \dvi doesn't work \letThereBe{\rnd}{1}{\mathup{#1}} %random variable \letThereBe{\vr}{1}{\mathbf{#1}} %random vector or matrix \letThereBe{\vrr}{1}{\boldsymbol{#1}} %random vector if \vr doesn't work \letThereBe{\dvr}{1}{\vr{\dot{#1}}} %differentiated vector or matrix \letThereBe{\vb}{1}{\pmb{#1}} %#TODO \letThereBe{\dvb}{1}{\vb{\dot{#1}}} %#TODO \letThereBe{\oper}{1}{\mathsf{#1}} \letThereBe{\quotient}{2}{{^{\displaystyle #1}}/{_{\displaystyle #2}}} % Basic notation - general \letThereBe{\set}{1}{\left\{#1\right\}} \letThereBe{\seqnc}{4}{\set{#1_{#2}}_{#2 = #3}^{#4}} \letThereBe{\Seqnc}{3}{\set{#1}_{#2}^{#3}} \letThereBe{\brackets}{1}{\left( #1 \right)} \letThereBe{\parentheses}{1}{\left[ #1 \right]} \letThereBe{\dom}{1}{\mcal{D}\, \brackets{#1}} \letThereBe{\complexConj}{1}{\overline{#1}} \LetThereBe{\divider}{\; \vert \;} \LetThereBe{\gets}{\leftarrow} \letThereBe{\rcases}{1}{\left.\begin{aligned}#1\end{aligned}\right\}} \letThereBe{\rcasesAt}{2}{\left.\begin{alignedat}{#1}#2\end{alignedat}\right\}} \letThereBe{\lcases}{1}{\begin{cases}#1\end{cases}} \letThereBe{\lcasesAt}{2}{\left\{\begin{alignedat}{#1}#2\end{alignedat}\right.} \letThereBe{\evaluateAt}{2}{\left.#1\right|_{#2}} \LetThereBe{\Mod}{\;\mathrm{mod}\;} \LetThereBe{\bigO}{O} \letThereBe{\BigO}{1}{\bigO\brackets{#1}} % Special symbols \LetThereBe{\const}{\mathrm{const}} \LetThereBe{\konst}{\mathrm{konst.}} \LetThereBe{\vf}{\varphi} \LetThereBe{\ve}{\varepsilon} \LetThereBe{\tht}{\theta} \LetThereBe{\Tht}{\Theta} \LetThereBe{\after}{\circ} \LetThereBe{\lmbd}{\lambda} \LetThereBe{\Lmbd}{\Lambda} % Shorthands \LetThereBe{\xx}{\vi x} \LetThereBe{\yy}{\vi y} \LetThereBe{\XX}{\vi X} \LetThereBe{\AA}{\vi A} \LetThereBe{\bb}{\vi b} \LetThereBe{\vvf}{\vi \vf} \LetThereBe{\ff}{\vi f} \LetThereBe{\gg}{\vi g} % Basic functions \letThereBe{\absval}{1}{\left| #1 \right|} \LetThereBe{\id}{\mathrm{id}} \letThereBe{\floor}{1}{\left\lfloor #1 \right\rfloor} \letThereBe{\ceil}{1}{\left\lceil #1 \right\rceil} \declareMathematics{\image}{im} %image \declareMathematics{\domain}{dom} %image \declareMathematics{\tg}{tg} \declareMathematics{\sign}{sign} \declareMathematics{\card}{card} %cardinality \letThereBe{\setSize}{1}{\left| #1 \right|} \LetThereBe{\countElements}{\#} \declareMathematics{\exp}{exp} \letThereBe{\Exp}{1}{\exp\brackets{#1}} \LetThereBe{\ee}{\mathrm{e}} \letThereBe{\indicator}{1}{\mathbb{I}_{#1}} \declareMathematics{\arccot}{arccot} \declareMathematics{\gcd}{gcd} % Greatest Common Divisor \declareMathematics{\lcm}{lcm} % Least Common Multiple \letThereBe{\limInfty}{1}{\lim_{#1 \to \infty}} \letThereBe{\limInftyM}{1}{\lim_{#1 \to -\infty}} % Useful commands \letThereBe{\onTop}{2}{\mathrel{\overset{#2}{#1}}} \letThereBe{\onBottom}{2}{\mathrel{\underset{#2}{#1}}} \letThereBe{\tOnTop}{2}{\mathrel{\overset{\text{#2}}{#1}}} \letThereBe{\tOnBottom}{2}{\mathrel{\underset{\text{#2}}{#1}}} \LetThereBe{\EQ}{\onTop{=}{!}} \LetThereBe{\letDef}{:=} %#TODO: change the symbol \LetThereBe{\isPDef}{\onTop{\succ}{?}} \LetThereBe{\inductionStep}{\tOnTop{=}{induct. step}} \LetThereBe{\fromDef}{\triangleq} % Optimization \declareMathematicsStar{\argmin}{argmin} \declareMathematicsStar{\argmax}{argmax} \letThereBe{\maxOf}{1}{\max\set{#1}} \letThereBe{\minOf}{1}{\min\set{#1}} \declareMathematics{\prox}{prox} \declareMathematics{\loss}{loss} \declareMathematics{\supp}{supp} \letThereBe{\Supp}{1}{\supp\brackets{#1}} \LetThereBe{\constraint}{\text{s.t.}\;} $$ $$ % Operators - Analysis \LetThereBe{\hess}{\nabla^2} \LetThereBe{\lagr}{\mcal L} \LetThereBe{\lapl}{\Delta} \declareMathematics{\grad}{grad} \declareMathematics{\divergence}{div} \declareMathematics{\Dgrad}{D} \LetThereBe{\gradient}{\nabla} \LetThereBe{\jacobi}{\nabla} \LetThereBe{\Jacobi}{\vi{\mathrm J}} \letThereBe{\jacobian}{2}{\Dgrad_{#1}\brackets{#2}} \LetThereBe{\d}{\mathrm{d}} \LetThereBe{\dd}{\,\mathrm{d}} \letThereBe{\partialDeriv}{2}{\frac {\partial #1} {\partial #2}} \letThereBe{\npartialDeriv}{3}{\partialDeriv{^{#1} #2} {#3^{#1}}} \letThereBe{\partialOp}{1}{\frac {\partial} {\partial #1}} \letThereBe{\npartialOp}{2}{\frac {\partial^{#1}} {\partial #2^{#1}}} \letThereBe{\pDeriv}{2}{\partialDeriv{#1}{#2}} \letThereBe{\npDeriv}{3}{\npartialDeriv{#1}{#2}{#3}} \letThereBe{\deriv}{2}{\frac {\d #1} {\d #2}} \letThereBe{\nderiv}{3}{\frac {\d^{#1} #2} {\d #3^{#1}}} \letThereBe{\derivOp}{1}{\frac {\d} {\d #1}\,} \letThereBe{\nderivOp}{2}{\frac {\d^{#1}} {\d #2^{#1}}\,} % Convergence \LetThereBe{\pointwiseTo}{\to} \LetThereBe{\uniformlyTo}{\rightrightarrows} \LetThereBe{\normallyTo}{\tOnTop{\longrightarrow}{norm}} \LetThereBe{\compactlyTo}{\tOnTop{\longrightarrow}{comp.}} \LetThereBe{\locallyUnifTo}{\tOnTop{\longrightarrow}{l.u.}} % Curves \letThereBe{\graphOf}{1}{\parentheses{#1}} \declareMathematics{\interior}{Int} % complex \LetThereBe{\Cinfty}{\tilde{\C}} \declareMathematics{\residual}{res} \letThereBe{\resAt}{1}{\residual_{#1}} \declareMathematics{\complexarg}{arg} \declareMathematics{\complexArg}{Arg} \LetThereBe{\carg}{\complexarg} \LetThereBe{\cArg}{\complexArg} \LetThereBe{\IM}{\mathfrak{Im}} \LetThereBe{\RE}{\mathfrak{Re}} \letThereBe{\imOf}{1}{\IM\,#1} \letThereBe{\reOf}{1}{\RE\,#1} \letThereBe{\ImOf}{1}{\IM \brackets{#1}} \letThereBe{\ReOf}{1}{\RE \brackets{#1}} $$ $$ % Linear algebra \letThereBe{\norm}{1}{\left\lVert #1 \right\rVert} \letThereBe{\seminorm}{1}{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert #1 \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}} \letThereBe{\scal}{2}{\left\langle #1, #2 \right\rangle} \letThereBe{\avg}{1}{\overline{#1}} \letThereBe{\Avg}{1}{\bar{#1}} \letThereBe{\linspace}{1}{\mathrm{lin}\set{#1}} \letThereBe{\algMult}{1}{\mu_{\mathrm A} \brackets{#1}} \letThereBe{\geomMult}{1}{\mu_{\mathrm G} \brackets{#1}} \LetThereBe{\Nullity}{\mathrm{nullity}} \letThereBe{\nullity}{1}{\Nullity \brackets{#1}} \LetThereBe{\nulty}{\nu} \declareMathematics{\SpanOf}{span} \letThereBe{\Span}{1}{\SpanOf\set{#1}} \LetThereBe{\projection}{\Pi} % Linear algebra - Matrices \LetThereBe{\tr}{\top} \LetThereBe{\Tr}{^\tr} \LetThereBe{\pinv}{\dagger} \LetThereBe{\Pinv}{^\dagger} \LetThereBe{\Inv}{^{-1}} \LetThereBe{\ident}{\vi{I}} \letThereBe{\mtr}{1}{\begin{pmatrix}#1\end{pmatrix}} \letThereBe{\bmtr}{1}{\begin{bmatrix}#1\end{bmatrix}} \declareMathematics{\trace}{tr} \declareMathematics{\diagonal}{diag} \declareMathematics{\rank}{rank} % Multilinear algebra \LetThereBe{\tensorProd}{\otimes} \LetThereBe{\tprod}{\tensorProd} \LetThereBe{\extProd}{\wedge} \LetThereBe{\wdg}{\extProd} \LetThereBe{\wedges}{\wedge \dots \wedge} \declareMathematics{\altMap}{Alt} $$ $$ % Statistics \LetThereBe{\iid}{\overset{\text{i.i.d.}}{\sim}} \LetThereBe{\ind}{\overset{\text{ind}}{\sim}} \LetThereBe{\condp}{\,\vert\,} \letThereBe{\complementOf}{1}{{{#1}^c}} \LetThereBe{\acov}{\gamma} \LetThereBe{\acf}{\rho} \LetThereBe{\stdev}{\sigma} \LetThereBe{\procMean}{\mu} \LetThereBe{\procVar}{\stdev^2} \declareMathematics{\variance}{var} \letThereBe{\Variance}{1}{\variance \brackets{#1}} \declareMathematics{\cov}{cov} \declareMathematics{\corr}{cor} \letThereBe{\sampleVar}{1}{\rnd S^2_{#1}} \letThereBe{\populationVar}{1}{V_{#1}} \declareMathematics{\expectedValue}{\mathbb{E}} \declareMathematics{\rndMode}{Mode} \letThereBe{\RndMode}{1}{\rndMode\brackets{#1}} \letThereBe{\expect}{1}{\expectedValue #1} \letThereBe{\Expect}{1}{\expectedValue \brackets{#1}} \letThereBe{\expectIn}{2}{\expectedValue_{#1} #2} \letThereBe{\ExpectIn}{2}{\expectedValue_{#1} \brackets{#2}} \LetThereBe{\betaF}{\mathrm B} \LetThereBe{\fisherMat}{J} \LetThereBe{\mutInfo}{I} \LetThereBe{\expectedGain}{I_e} \letThereBe{\KLDiv}{2}{D\brackets{#1 \parallel #2}} \LetThereBe{\entropy}{H} \LetThereBe{\diffEntropy}{h} \LetThereBe{\probF}{\pi} \LetThereBe{\densF}{\vf} \LetThereBe{\att}{_t} %at time \letThereBe{\estim}{1}{\hat{#1}} \letThereBe{\estimML}{1}{\hat{#1}_{\mathrm{ML}}} \letThereBe{\estimOLS}{1}{\hat{#1}_{\mathrm{OLS}}} \letThereBe{\estimMAP}{1}{\hat{#1}_{\mathrm{MAP}}} \letThereBe{\predict}{3}{\estim {\rnd #1}_{#2 | #3}} \letThereBe{\periodPart}{3}{#1+#2-\ceil{#2/#3}#3} \letThereBe{\infEstim}{1}{\tilde{#1}} \letThereBe{\predictDist}{1}{{#1}^*} \LetThereBe{\backs}{\oper B} \LetThereBe{\diff}{\oper \Delta} \LetThereBe{\BLP}{\oper P} \LetThereBe{\arPoly}{\Phi} \letThereBe{\ArPoly}{1}{\arPoly\brackets{#1}} \LetThereBe{\maPoly}{\Theta} \letThereBe{\MaPoly}{1}{\maPoly\brackets{#1}} \letThereBe{\ARmod}{1}{\mathrm{AR}\brackets{#1}} \letThereBe{\MAmod}{1}{\mathrm{MA}\brackets{#1}} \letThereBe{\ARMA}{2}{\mathrm{ARMA}\brackets{#1, #2}} \letThereBe{\sARMA}{3}{\mathrm{ARMA}\brackets{#1}\brackets{#2}_{#3}} \letThereBe{\SARIMA}{3}{\mathrm{ARIMA}\brackets{#1}\brackets{#2}_{#3}} \letThereBe{\ARIMA}{3}{\mathrm{ARIMA}\brackets{#1, #2, #3}} \LetThereBe{\pacf}{\alpha} \letThereBe{\parcorr}{3}{\rho_{#1 #2 | #3}} \LetThereBe{\noise}{\mathscr{N}} \LetThereBe{\jeffreys}{\mathcal J} \LetThereBe{\likely}{\mcal L} \letThereBe{\Likely}{1}{\likely\brackets{#1}} \LetThereBe{\loglikely}{\mcal l} \letThereBe{\Loglikely}{1}{\loglikely \brackets{#1}} \LetThereBe{\CovMat}{\Gamma} \LetThereBe{\covMat}{\vi \CovMat} \LetThereBe{\rcovMat}{\vrr \CovMat} \LetThereBe{\AIC}{\mathrm{AIC}} \LetThereBe{\BIC}{\mathrm{BIC}} \LetThereBe{\AICc}{\mathrm{AIC}_c} \LetThereBe{\nullHypo}{H_0} \LetThereBe{\altHypo}{H_1} \LetThereBe{\rve}{\rnd \ve} \LetThereBe{\rtht}{\rnd \theta} \LetThereBe{\rX}{\rnd X} \LetThereBe{\rY}{\rnd Y} \LetThereBe{\rZ}{\rnd Z} \LetThereBe{\rA}{\rnd A} \LetThereBe{\rB}{\rnd B} \LetThereBe{\vrZ}{\vr Z} \LetThereBe{\vrY}{\vr Y} \LetThereBe{\vrX}{\vr X} \LetThereBe{\rW}{\rnd W} \LetThereBe{\rS}{\rnd S} \LetThereBe{\rM}{\rnd M} \LetThereBe{\rtau}{\rnd \tau} % Bayesian inference \LetThereBe{\paramSet}{\mcal T} \LetThereBe{\sampleSet}{\mcal Y} \LetThereBe{\bayesSigmaAlg}{\mcal B} % Different types of convergence \LetThereBe{\inDist}{\onTop{\to}{d}} \letThereBe{\inDistWhen}{1}{\onBottom{\onTop{\longrightarrow}{d}}{#1}} \LetThereBe{\inProb}{\onTop{\to}{P}} \letThereBe{\inProbWhen}{1}{\onBottom{\onTop{\longrightarrow}{P}}{#1}} \LetThereBe{\inMeanSq}{\onTop{\to}{\ltwo}} \LetThereBe{\inltwo}{\onTop{\to}{\ltwo}} \letThereBe{\inMeanSqWhen}{1}{\onBottom{\onTop{\longrightarrow}{\ltwo}}{#1}} \LetThereBe{\convergeAS}{\tOnTop{\to}{a.s.}} \letThereBe{\convergeASWhen}{1}{\onBottom{\tOnTop{\longrightarrow}{a.s.}}{#1}} % Asymptotic qualities \LetThereBe{\simAsymp}{\tOnTop{\sim}{as.}} % Stochastic analysis \letThereBe{\diffOn}{2}{\diff #1_{[#2]}} % \LetThereBe{\timeSet}{\Theta} \LetThereBe{\eventSet}{\Omega} \LetThereBe{\filtration}{\mcal F} % TODO: Rename allFiltrations and the like \letThereBe{\allFiltrations}{1}{\set{\filtration_t}_{#1}} \letThereBe{\natFilter}{1}{\filtration_t^{#1}} \letThereBe{\NatFilter}{2}{\filtration_{#2}^{#1}} \letThereBe{\filterAll}{1}{\set{#1}_{t \geq 0}} \letThereBe{\FilterAll}{2}{\set{#1}_{#2}} \LetThereBe{\borelAlgebra}{\mcal B} \LetThereBe{\sAlgebra}{\mcal A} \LetThereBe{\quadVar}{Q} \LetThereBe{\totalVar}{V} \LetThereBe{\adaptIntProcs}{\mcal M} \letThereBe{\reflectProc}{2}{#1^{#2}} $$ $$ % Distributions \letThereBe{\WN}{2}{\mathrm{WN}\brackets{#1,#2}} \declareMathematics{\uniform}{Unif} \declareMathematics{\binomDist}{Bi} \declareMathematics{\negbinomDist}{NBi} \declareMathematics{\betaDist}{Beta} \declareMathematics{\betabinomDist}{BetaBin} \declareMathematics{\gammaDist}{Gamma} \declareMathematics{\igammaDist}{IGamma} \declareMathematics{\invgammaDist}{IGamma} \declareMathematics{\expDist}{Ex} \declareMathematics{\poisDist}{Po} \declareMathematics{\erlangDist}{Er} \declareMathematics{\altDist}{A} \declareMathematics{\geomDist}{Ge} \LetThereBe{\normalDist}{\mathcal N} %\declareMathematics{\normalDist}{N} \letThereBe{\normalD}{1}{\normalDist \brackets{#1}} \letThereBe{\mvnormalD}{2}{\normalDist_{#1} \brackets{#2}} \letThereBe{\NormalD}{2}{\normalDist \brackets{#1, #2}} \LetThereBe{\lognormalDist}{\log\normalDist} $$ $$ % Game Theory \LetThereBe{\doms}{\succ} \LetThereBe{\isdom}{\prec} \letThereBe{\OfOthers}{1}{_{-#1}} \LetThereBe{\ofOthers}{\OfOthers{i}} \LetThereBe{\pdist}{\sigma} \letThereBe{\domGame}{1}{G_{DS}^{#1}} \letThereBe{\ratGame}{1}{G_{Rat}^{#1}} \letThereBe{\bestRep}{2}{\mathrm{BR}_{#1}\brackets{#2}} \letThereBe{\perf}{1}{{#1}_{\mathrm{perf}}} \LetThereBe{\perfG}{\perf{G}} \letThereBe{\imperf}{1}{{#1}_{\mathrm{imp}}} \LetThereBe{\imperfG}{\imperf{G}} \letThereBe{\proper}{1}{{#1}_{\mathrm{proper}}} \letThereBe{\finrep}{2}{{#2}_{#1{\text -}\mathrm{rep}}} %T-stage game \letThereBe{\infrep}{1}{#1_{\mathrm{irep}}} \LetThereBe{\repstr}{\tau} %strategy in a repeated game \LetThereBe{\emptyhist}{\epsilon} \letThereBe{\extrep}{1}{{#1^{\mathrm{rep}}}} \letThereBe{\avgpay}{1}{#1^{\mathrm{avg}}} \LetThereBe{\succf}{\pi} %successor function \LetThereBe{\playf}{\rho} %player function \LetThereBe{\actf}{\chi} %action function $$ $$ \LetThereBe{\fourierOp}{\mcal{F}} \letThereBe{\fourier}{1}{\widehat{#1}} \letThereBe{\ifourier}{1}{\check{#1}} % Shortcuts \letThereBe{\FT}{1}{\fourier{#1}} \letThereBe{\iFT}{1}{\ifourier{#1}} \LetThereBe{\FTOp}{\fourierOp} \LetThereBe{\lspace}{\mcal L} \LetThereBe{\lone}{\lspace^{1}} \letThereBe{\Lone}{1}{\lone\brackets{#1}} \LetThereBe{\ltwo}{\lspace^2} \letThereBe{\Ltwo}{1}{\ltwo\brackets{#1}} \letThereBe{\lp}{1}{\lspace^{#1}} \letThereBe{\Lp}{2}{\lp{#1}\brackets{#2}} \LetThereBe{\linfty}{\lspace^{\infty}} \letThereBe{\Linfty}{1}{\linfty\brackets{#1}} \LetThereBe{\ltwoEq}{\onTop{=}{\ltwo}} \letThereBe{\decayContf}{1}{\mcal C_0\brackets{#1}} \letThereBe{\cinftyContf}{1}{\mcal C^{\infty}_c\brackets{#1}} \LetThereBe{\unitBall}{\mathbb{S}} \LetThereBe{\unitBallSurface}{S} \LetThereBe{\gammaF}{\Gamma} \letThereBe{\GammaF}{1}{\gammaF\brackets{#1}} \LetThereBe{\betaF}{B} \letThereBe{\BetaF}{1}{\betaF\brackets{#1}} \LetThereBe{\ofOrder}{\asymp} \declareMathematics{\vol}{vol} \LetThereBe{\holomorphic}{\mcal H} \LetThereBe{\hlmr}{\holomorphic} \LetThereBe{\schwartz}{\mcal S} \LetThereBe{\testSpace}{\mcal D} \letThereBe{\TestSpace}{1}{\testSpace\brackets{#1}} \LetThereBe{\bumpf}{\psi} \letThereBe{\Bumpf}{1}{\bumpf\brackets{#1}} \letThereBe{\reverse}{1}{\check{#1}} \LetThereBe{\translationOp}{\mcal{T}} \letThereBe{\translateBy}{1}{\translationOp_{#1}} \LetThereBe{\flcOp}{\mathscr{F}} \letThereBe{\flc}{1}{\flcOp\brackets{#1}} \LetThereBe{\hodge}{*} \LetThereBe{\tangent}{\mathrm{T}} \letThereBe{\tangentAt}{1}{\tangent_{#1}} \letThereBe{\lieBracket}{2}{\left[#1, #2\right]} \declareMathematics{\degree}{deg} \LetThereBe{\forms}{\mathscr{F}} \letThereBe{\kforms}{1}{\forms^{#1}} \letThereBe{\formsOver}{1}{\forms\brackets{#1}} \letThereBe{\kformsOver}{2}{\kforms{#1}\brackets{#2}} \letThereBe{\pullback}{1}{{#1}^*} \LetThereBe{\atlas}{\mcal A} \LetThereBe{\adjd}{\oper \delta} \LetThereBe{\connect}{\nabla} \LetThereBe{\christoffel}{\Gamma} \letThereBe{\lengthOf}{1}{\mathcal{l}\brackets{#1}} \LetThereBe{\HOT}{\mathrm{h.o.t.}} $$ $$ % ODEs \LetThereBe{\timeInt}{\mcal I} \LetThereBe{\stimeInt}{\mcal J} \LetThereBe{\Wronsk}{\mcal W} \letThereBe{\wronsk}{1}{\Wronsk \parentheses{#1}} \LetThereBe{\prufRadius}{\rho} \LetThereBe{\prufAngle}{\vf} \LetThereBe{\weyr}{\sigma} \LetThereBe{\linDifOp}{\mathsf{L}} \LetThereBe{\Hurwitz}{\vi H} \letThereBe{\hurwitz}{1}{\Hurwitz \brackets{#1}} % Cont. Models \LetThereBe{\dirac}{\delta} \LetThereBe{\torus}{\mathbb{T}} % PDEs % \avint -- defined in format-respective tex files \LetThereBe{\fundamental}{\Phi} \LetThereBe{\fund}{\fundamental} \letThereBe{\normaDeriv}{1}{\partialDeriv{#1}{\vec{n}}} \letThereBe{\volAvg}{2}{\avint_{\ball{#1}{#2}}} \LetThereBe{\VolAvg}{\volAvg{x}{\ve}} \letThereBe{\surfAvg}{2}{\avint_{\boundaryOf \ball{#1}{#2}}} \LetThereBe{\SurfAvg}{\surfAvg{x}{\ve}} \LetThereBe{\corrF}{\varphi^{\times}} \LetThereBe{\greenF}{G} \letThereBe{\reflect}{1}{\tilde{#1}} \LetThereBe{\conv}{*} \letThereBe{\dotP}{2}{#1 \cdot #2} \letThereBe{\translation}{1}{\tau_{#1}} \declareMathematics{\dist}{dist} \letThereBe{\regularizef}{1}{\eta_{#1}} \letThereBe{\fourier}{1}{\widehat{#1}} \letThereBe{\ifourier}{1}{\check{#1}} \LetThereBe{\fourierOp}{\mcal F} \LetThereBe{\ifourierOp}{\mcal F^{-1}} \letThereBe{\FourierOp}{1}{\fourierOp\set{#1}} \letThereBe{\iFourierOp}{1}{\ifourierOp\set{#1}} \LetThereBe{\laplaceOp}{\mcal L} \letThereBe{\LaplaceOp}{1}{\laplaceOp\set{#1}} \letThereBe{\Norm}{1}{\absval{#1}} % SINDy \LetThereBe{\Koop}{\mcal K} \letThereBe{\oneToN}{1}{\left[#1\right]} \LetThereBe{\meas}{\mathrm{m}} \LetThereBe{\stateLoss}{\mcal J} \LetThereBe{\lagrm}{p} % Stochastic analysis \LetThereBe{\RiemannInt}{(\mcal R)} \LetThereBe{\RiemannStieltjesInt}{(\mcal {R_S})} \LetThereBe{\LebesgueInt}{(\mcal L)} \LetThereBe{\ItoInt}{(\mcal I)} \LetThereBe{\Stratonovich}{\circ} \LetThereBe{\infMean}{\alpha} \LetThereBe{\infVar}{\beta} % Dynamical systems \LetThereBe{\nUnit}{\mathrm N} \LetThereBe{\timeUnit}{\mathrm T} % Masters thesis \LetThereBe{\evolOp}{\oper{\vf}} \letThereBe{\obj}{1}{\mathbb{#1}} \LetThereBe{\timeSet}{\obj T} \LetThereBe{\stateSpace}{\obj X} \LetThereBe{\contStateSpace}{\stateSpace_{C}} \LetThereBe{\orbit}{Or} \letThereBe{\Orbit}{1}{\orbit\brackets{#1}} \LetThereBe{\limitSet}{\obj \Lambda} \LetThereBe{\crossSection}{\obj \Sigma} \declareMathematics{\codim}{codim} % Left and right closed-or-open intervals \LetThereBe{\lco}{\langle} \LetThereBe{\rco}{\rangle} \letThereBe{\testInt}{1}{\mathrm{Int}_{#1}} \letThereBe{\evalOp}{1}{\oper{\eta}_{#1}} \LetThereBe{\nonzeroEl}{\bullet} \LetThereBe{\zeroEl}{\circ} \LetThereBe{\solOp}{\oper{S}} \LetThereBe{\infGen}{\oper{A}} \LetThereBe{\indexSet}{\mcal I} \letThereBe{\indicesOf}{1}{\indexSet\parentheses{#1}} \letThereBe{\IndicesOf}{2}{\indexSet_{#2}\parentheses{#1}} \LetThereBe{\meshGrid}{\obj M} \declareMathematics{\starter}{starter} \declareMathematics{\indexer}{indx} \declareMathematics{\enumerator}{enum} \LetThereBe{\inSS}{_{\infty}} \LetThereBe{\manifold}{\mcal M} \LetThereBe{\curve}{\mcal C} % Numerical methods \declareMathematics{\globErr}{err} \declareMathematics{\locErr}{le} \declareMathematics{\locTrErr}{lte} \declareMathematics{\estimErr}{est} \declareMathematics{\incrementFunc}{Inc} \letThereBe{\incrementF}{1}{\incrementFunc \brackets{#1}} \LetThereBe{\discreteNodes}{\mcal T} \LetThereBe{\stableFunc}{R} \letThereBe{\stableF}{1}{\stableFunc\brackets{#1}} \LetThereBe{\stableRegion}{\Omega} %Stochastic analysis \LetThereBe{\RiemannInt}{(\mcal R)} \LetThereBe{\RiemannStieltjesInt}{(\mcal {R_S})} \LetThereBe{\LebesgueInt}{(\mcal L)} \LetThereBe{\ItoInt}{(\mcal I)} \LetThereBe{\Stratonovich}{\circ} \LetThereBe{\infMean}{\alpha} \LetThereBe{\infVar}{\beta} %Optimization \LetThereBe{\goldRatio}{\tau} %Interpolation \LetThereBe{\lagrPoly}{l} $$

3.1 Introduction to Multilinear Algebra

Definition 3.1 (Multilinear map) Let \(V_1, \dots, V_k, V\) be vector spaces over a field \(\K\). A map \(m : V_1 \times \dots \times V_k \to V\) is called multilinear if it is linear in each coordinate, i.e., \[ m\brackets{\vi v_1, \dots, \vi v_{i-1}, \lmbd \vi v_i + \vi w, \vi v_{i+1}, \dots, \vi v_k} = \lmbd m(\vi v_1, \dots, \vi v_k) + m\brackets{\vi v_1, \dots, \vi v_{i - 1}, \vi w, \vi v_{i+1}, \dots, \vi v_k} \] for all \(i \in \oneToN{k}\) and every \(\lmbd \in \K\), and \(\vi w \in V_i\).

It turns out we can characterize, or more precisely decompose, multilinear functions rather nicely. To this end, we would like to construct a vector space \(W\) and a multilinear map \(\iota : V_1 \times \dots \times V_k \to W\) such that every multilinear map \(m : V_1 \times \dots \times V_k \to V\) can be decomposed into \(\iota\) and a linear map \(\vf\) with \(m = \vf \after \iota\), where \(\iota\) does not depend on \(m\), but \(\vf\) does – we call this (ability to always construct a) decomposition a universal property.

Figure 3.1: Decomposition of the multilinear map \(m\) into the \(m\)-independent part \(\iota\) and dependent part \(\vf\).

Crucially, we can make the following observation.

Proposition 3.1 If such a pair \((W, \iota)\), as described above conforming to the universal property, exists, then it is unique up to an isomorphism.

Proof. Indeed, as \(\iota\) does not depend on \(m\) it must map the space of all multilinear functions \(V_1 \times \dots \times V_k \to V\) to space of linear functions \(W \to V\) (if it was a subspace, we could take that as \(W\)). Moreover, \(\iota\) is also injective, as if there was \(\vi v = (\vi v_1, \dots, \vi v_k)\) such that \(\vi v \neq \vi 0\) but \(\iota(\vi v) = 0_W\), then for multilinear \(m\) such that \(m(\vi v) \neq 0_V\) we have \[ m(\vi v) = (\vf_m \after \iota)(\vi v) = \vf_m\brackets{\iota(\vi v)} = \vf_m\brackets{0_w} \tOnTop{=}{linear} 0_V, \] hence we have a contradiction. Thus \(\iota\) is a linear bijection between \(V_1 \times \dots \times V_K\) and \(W\), i.e., “various” \(W\)’s differ only up-to isomorphism.

Now we define that the tensor product \(\tprod\) for a \(k\)-linear function \(f(\vi x_{1, \dots,k})\) and a \(l\)-linear function \(g(\vi x_{k+1, \dots, k+l})\) as \[ (f \tprod g) (\vi x_{1, \dots, k+l}) = f(\vi x_{1, \dots, k})\cdot g(\vi x_{k+1, \dots, k+l}). \tag{3.1}\] In particular, we can take \(f,g\) be covectors of given vector spaces! Note that the tensor product is not necessarily associative, however it is distributive, i.e., in general \[ f \tprod g \neq g \tprod f \quad \& \quad (f \tprod g) \tprod h = f \tprod (g \tprod h). \]

3.1.1 Abstract Tensor Product of Vector Spaces

Firstly, let us note this section was added on top of the material of the lecture is mainly based on [1, chpt. 12.1.1].

So far, we have basically shown that multilinear functions may be viewed as the span of \(\vi v^1 \tprod \dots \tprod \vi v^k\), where crucially \(\vi v^1, \dots, \vi v^k\) are covectors (i.e., linear functionals on \(V_i\)). However this construction can also be extended to general finite-dimensional vector spaces.

Let us have vector spaces \(V_1, \dots, V_k\) over the field \(\K\). To define the vector product of these spaces, we first need to understand formal linear combinations. Intuitively, formal linear combinations of \(V(\K)\) are expressions of form \(\sum_{i = 1}^m a_i \vi v_i\) for \(a_i \in \K\) and \(\vi v_i \in V\) — however we may also define them for a general set \(S\) without the necessary underlying algebraic structure that would allow for multiplication by scalars and/or addition of these elements.

Definition 3.2 (Formal linear combination) For any set \(S\), a formal linear combination of elements of \(S\) is a function \(f : S \to \K\) such that \(f(s) = 0\) for all but finitely many \(s \in S\). Consequently, the free vector space on \(S\), denoted \(\flc{S}\), is the set of all formal linear combinations of elements of \(S\).

Intuitively, we can think of every \(f \in \flc{S}\) as “\(\sum_{i = 1}^n a_i s_i\)” (note that we cannot, a priori, add or multiply elements from \(S\)) such that \(f\) would return the scalar \(a_i\) for each \(s_i\) in the sum (and zero for all other elements of \(S\) not present).

Evidently, under point-wise addition (for \(f,g \in \flc{S}\) we have \((f+g)(s) = f(s) + g(s)\)) and scalar multiplication (for \(\lmbd \in K\) and \(f \in \flc{S}\) we have \(f(\lmbd s) = \lmbd f(s)\)), \(\flc{S}\) becomes a vector over \(\K\). Furthermore, for each element \(x \in S\), there is a function \(\dirac_x \in \flc{S}\) that takes the value \(1\) on \(x\) and zero on all other elements of \(S\), and we identify this function with \(x\) itself, thus \(S\) can be viewed as a subset of \(\flc{S}\). Every element \(f \in \flc{S}\) can then be uniquely written as \(f = \sum_{i = 1}^m a_i x_i\) where \(x_1, \dots, x_m\) are the elements of \(S\) for which \(f(x_i) \neq 0\), and \(a_i = f(x_i)\). In other words, \(S\) is a basis of \(\flc{S}\), which is thus finite-dimensional if and only if \(S\) is a finite set.

Proposition 3.2 For any set \(S\) and any vector space \(W\), every map \(\vi A : S \to W\) has a unique extension to a linear map \(\overline{\vi A} : \flc{S} \to W\).

Now we focus on \(V_1, \dots, V_k\) vector spaces over \(\K\). We form the free vector space \(\flc{V_1 \times \dots \times V_k}\) on their Cartesian product, which is the set of all formal linear combinations of \(k\)-tuples. Moreover, let \(\mcal R \subset \flc{V_1 \times \dots \times V_k}\) spanned by all elements of the following forms: \[ \begin{gathered} (\vi v_1, \dots, a \vi v_i, \dots, \vi v_k) - a (\vi v_1, \dots, \vi v_i, \dots, \vi v_k) \\ (\vi v_1, \dots, \vi v_i + \vi v_i', \dots, \vi v_k) - (\vi v_1, \dots, \vi v_i, \dots, \vi v_k) - (\vi v_1, \dots, \vi v_i', \dots, \vi v_k), \end{gathered} \tag{3.2}\] with \(\vi v_i, \vi v_i' \in V_i\) for all \(i \in \oneToN{k}\) and \(a \in \K\). We finally define the (abstract) tensor product of the spaces \(V_1, \dots, V_k\), denoted \(V_1 \tprod \dots \tprod V_k\) to be the following quotient space: \[ V_1 \tprod \dots \tprod V_k \letDef \quotient{\flc{V_1 \times \dots \times V_k}}{\mcal R}, \tag{3.3}\] and let \(\projection: \flc{V_1 \times \dots \times V_k} \to V_1 \tprod \dots \tprod V_k\) be the natural projection. Moreover, the equivalence class of an element \((\vi v_1, \dots, \vi v_k) \in V_1 \tprod \dots \tprod V_k\) is denoted \[ \vi v_1 \tprod \dots \tprod \vi v_k = \projection(\vi v_1, \dots, \vi v_k), \tag{3.4}\] and is called the (abstract) tensor of \(\vi v_1, \dots, \vi v_k\). It then follows from the definition of the tensor product \(V_1 \tprod \dots \tprod V_k\) that \(\vi v_1 \tprod \dots \tprod \vi v_k\) is multilinear in the sense of Definition 3.1. Intuitively, elements \((\vi v_1 + \vi v_1', \dots, \vi v_k)\) and \((\vi v_1, \dots, \vi v_k) + (\vi v_1', \dots, \vi v_k)\) are different symbols/elements \(\flc{V_1 \times \dots \times V_k}\) (both are valid as \(V_1,\dots, V_k\) are vector spaces, as well as \(\flc{S}\) for any set \(S\)), i.e., their “coefficients functions” might not return the same results. We “fix” this discrepancy by identifying them into one equivalence by \(\mcal R\), thus enforcing the multilinearity.

Using the observation Proposition 3.1 and the ideas from Section 3.1.1, we define \[ \begin{aligned} W &\letDef \Span{\vi v_1 \tprod \vi v_2 \tprod \dots \tprod \vi v_k \divider \vi v_1 \in V_1, \dots, \vi v_k \in V_k} \\ &= \Span{\vi b_{1j_1} \tprod \vi b_{2j_2} \tprod \dots \tprod \vi b_{kj_k} \divider \vi b_{i1}, \dots, \vi b_{i\dim V_i} \text{ the basis of } V_i}, \end{aligned} \tag{3.5}\] where \(W = V_1 \tprod V_2 \tprod \dots \tprod V_k\) in setting of Proposition 3.1. Note that elements \(\vi b_{1j_1} \tprod \vi b_{2j_2} \tprod \dots \tprod \vi b_{kj_k}\) form the basis of \(W\).

Remark 3.1 (Basis-free canonical objects). Let us explore turning a “supposedly” basis-dependent object into a canonical one. Let \(V\) be a vector space, then \(H(V, V)\) is the vector space endomorphism of \(V\), \[\begin{align*} H(V,V) &\letDef \set{\vf : V \to V \divider \vf \text{ is linear}} \\ &\Downarrow\\ H(V,V) &\cong V \tprod V^*, \end{align*}\] i.e., \(H(V,V)\) is isomorphic to vector space of elements \(v \tprod l : V \to V\) (where \(l \in V^*\) is a covector, i.e., linear functional), hence \(w \mapsto l(w) \cdot v\). For illustration, let \(V = \R^n\) with basis \(\vi e_{1}, \dots, \vi e_n\), then to each linear mapping \(\vf : \R^n \to \R^n\) there uniquely corresponds a matrix \(\vi A = (a_{ij})_{i,j = 1}^n \in \R^{n \times n}\), for which thus holds (per (3.5)) \[ \vi A \vi x = \sum_{i,j = 1}^n a_{ij} (\overbrace{\vi e_i}^{\in V} \tprod \overbrace{\vi e^j}^{\in V^*})(\vi x) = \sum_{i,j = 1}^n a_{ij} \vi e^j(\vi x) \vi e_i = \sum_{i, j = 1}^n a_{ij} \scal{\vi e_j}{\vi x} \vi e_i. \]

Now consider the trace \(\trace : V \tprod V^* \to \K\) with \(v \tprod l \mapsto l(v)\). Returning back to the special case \(V = \R^n\), we see \[ \trace (\vi A) = \sum_{i,j = 1}^n a_{ij} \vi e^j(\vi e_i) = \sum_{i,j = 1}^n a_{ij} \underbrace{\scal {\vi e_j} {\vi e_i}}_{\dirac_{i,j}} = \sum_{i = 1}^n a_{ii}. \] In other words, we have prescribed the trace without the employing any base, which was to be done.

3.1.2 Exterior Product of Vector Spaces

Definition 3.3 (Alternating map) Let \(V, Y\) be a vector spaces. A multilinear map \(m : V^k \to Y\) is called alternating if¹ \[ m(\vi v_1, \dots, \vi v_k) = \sign(\pi) m\brackets{\vi v_{\pi(1)}, \dots, \vi v_{\pi(k)}} \quad \forall \pi \in S_k. \]

Remark 3.2. Such maps can only be non-trivial if \(k \leq \dim V\). In other case, we can use [1, lemma 14.1.] and calculate several linear combinations to obtain \(m = -m\) (by the pigeon hole principle).

Remark 3.3. If \(k = \dim V\), \(m\) is uniquely defined by its value at \(\vi b_1, \dots, \vi b_k\) (basis). Then, \(m(\vi v_1, \dots, \vi v_k) = \det(\vi v_1, \dots, \vi v_k) \cdot \vi y\) for some \(\vi y \in Y\)

Definition 3.4 (Exterior (wedge) product) For a vector space \(V\) we define \[ \ExtProd^k V \letDef \Span{\vi v_1 \wdg \vi v_2 \wdg \dots \wdg \vi v_k \divider \vi v_i \in V} \] the \(k\)-th exterior power of \(V\) as the alternating analogue of the tensor product. Note the convention \(\ExtProd^0 V = \K\) for a vector space \(V\) over the field \(\K\). To this end, we set \[ \altMap : \vi v_1 \tprod \dots \tprod \vi v_k \mapsto \frac 1 {k!} \sum_{\pi \in S_k} \sign(\pi) (\vi v_{\pi(1)} \tprod \dots \tprod \vi v_{\pi(k)}), \] then \(\ExtProd^k V = \altMap (\tprod^k V)\). Alternatively, we take the perspective \[ \ExtProd^k V \cong \quotient{\tprod^k V}{\Span{\vi v_1 \tprod \dots \tprod \vi v_k \divider \vi v_i = \vi v_j \text{ for some } i \neq j}}. \tag{3.6}\] Intuitively, \(\altMap\) is the alternating variant of \(\iota\) from the introduction, and in (3.6) we simply “mod it out”.

Remark 3.4. Let \(\vi v_1, \dots, \vi v_k \in V\), then \(\vi v_1, \dots, \vi v_k\) are linear independent if and only if \(\vi v_1 \wdg \dots \wdg \vi v_k \neq 0\). One can think of this as an analogue for determinant, which also works for \(k < \dim V\).

Remark 3.5. If \(\vi v_1, \dots, \vi v_k \in V\) are linearly independent, then \[ \set{\vi x \in V \divider \vi v_1 \wdg \dots \wdg \vi v_k \wdg \vi x = 0} = \Span{\vi v_1, \dots, \vi v_k}, \] as per Remark 3.4 it is the set of all linearly dependent vectors in \(V\).

Example 3.1 Let \(\vi v_1, \vi v_2 \in \R^3\) be linearly independent, then we can define the plane spanned by \(\vi v_1, \vi v_2\) in the following ways: \[\begin{align*} \vi v_1 \wdg \vi v_2 \wdg \vi x &= 0, \\ \det(\vi v_1, \vi v_2, \vi x) &= 0, \\ \scal{\vi v_1 \times \vi v_2}{x} &= 0. \end{align*}\] Similarly, \(\vi v \wdg \vi x = 0\) describes a line, and for basis-representations \(\vi v = (a,b,c)\), \(\vi x = (x,y,z)\) we have that \[\begin{align*} \vi v \wdg \vi x &= (a \vi e_1 + b \vi e_2 + c \vi e_3) \wdg (x \vi e_1 + y \vi e_2 + z \vi e_3) \\ &= (ay - bx) (\vi e_1 \wdg \vi e_2) + (bz - cy) (\vi e_2 \wdg \vi e_3) + (cx - az) (\vi e_3 \wdg \vi e_1) \\ &\Downarrow\\ \vi v \wdg \vi x &= 0 \implies \lcases{ ay - bx = 0, \\ bz - cy = 0, \\ cz -a z = 0, } \end{align*}\] i.e., we get a system of linear equations.

Remark 3.6. Not every element of \(\ExtProd^k V\) is a (wedge-)product!

Proposition 3.3 Let \(\vi v \in \ExtProd^k V \neq 0_{\ExtProd^k V}\). Then \(\vi v\) can be expressed as (a scalar multiple of) \(\vi v_1 \wdg \dots \wdg \vi v_k\) if and only if the space \(L = \set{\vi x \in V \divider \vi v \wdg \vi x = 0}\) is \(k\)-dimensional.

For the proof of this proposition, we will need the following observation.

Lemma 3.1 If for \(\vi z \in \ExtProd^k V\) holds \(\vi z \wdg \vi w = 0\) for any \(\vi w \in \ExtProd^m V\), where \(k + m \leq n\), then \(\vi z = \vi 0\).

Proof. Let \(\vi v_1, \dots, \vi v_n \in V\) be its basis, then recall \(\vi z = \sum_{j_1 < \dots j_k} a_{j_1, \dots, j_k} \vi v_{j_1} \wdg \dots \wdg \vi v_{j_k} \in \ExtProd^k V\). If \(k + m \leq n\) and we pick an arbitrary basis vector \(\vi w\) of \(\ExtProd^{m} V\), i.e., \(\vi w = \vi v_{l_1} \wdg \dots \wdg \vi v_{l_m}\) with \(l_1 < \dots < l_m\), then \[ \vi z \wdg \vi w = \sum_{j_1 < \dots < j_k} a_{j_1, \dots, j_k} \vi v_{j_1} \wdg \dots \wdg \vi v_{j_k} \wdg \vi v_{l_1} \wdg \dots \wdg \vi v_{l_m}. \] Every term in this sum is either zero or a basis element of \(\ExtProd^{k+m} V\), which only occurs once (for only one choice of \(\vi w\) among the basis vectors of \(\ExtProd^k V\)). However, as it has to be identically zero for all, this means that \(a_{j_1, \dots, j_k}\) must be \(0\) if the given term is a basis vector of \(\ExtProd^{k+m} V\). Repeating over all choices of \(\vi w\) yields \(\vi z = \vi 0\).

Proof. Let us now prove Proposition 3.3.

“\(\implies\)”: From \(\vi 0 \neq \vi v = \vi v_1 \wdg \dots \wdg \vi v_k\) we get linear independence of \(\vi v_1, \dots, \vi v_k\) by Remark 3.4, thus their span (by Remark 3.5) is \(k\)-dimensional.

“\(\impliedby\)”: If \(L\) is \(k\)-dimensional, it may be written as a span of linearly independent vectors \(\vi v_1, \dots, \vi v_k\), i.e., \(L = \Span{\vi v_1, \dots \vi v_k}\), such that they can be extended to form the basis \(V\), namely \(\vi v_1, \dots, \vi v_k, \vi v_{k+1}, \dots, \vi v_n\). Now choose \(c\) such that \[ (\vi v - c \vi v_1 \wdg \cdots \wdg \vi v_k) \wdg \vi v_{k+1} \wdg \dots \wdg \vi v_n = \vi 0, \] i.e., \(\vi v \wdg \vi v_{k+1} \wdg \dots \wdg \vi v_n = c \vi v_1 \wdg \dots \wdg \vi v_n\). This is possible since both sides of the equality belong in \(\ExtProd^n V\), which is 1-dimensional, thus the quantities are proportional. Now take any \(\vi w \in \ExtProd^{n-k} V\) and consider \(\vi u = (\vi v - c \vi v_1 \wdg \dots \wdg \vi v_k) \wdg \vi w\). Without loss of generality, we can consider \(\vi w\) to be a basis element of \[ \ExtProd^{n-k} V = \Span{\vi v_{j_1} \wdg \dots \wdg \vi v_{j_{n-k}} \divider \vi v_i \in V \text{ and } j_1, \dots, j_{n-k} \text{ pairwise distinct}}. \] Now there are two possible cases for the form of \(\vi w\):

1. there exists some \(j_l \leq k\), and without loss of generality, \(l = 1\) \(\implies\) \(\vi u = \vi 0\) by the definition of \(L\);
2. all \(j_l > k\), then all \(j_l\) are exactly \(k+1, \dots, n\) and, once again without loss of generality, we can also assume in that order \(\implies\) by the definition of \(c\), it yields \(\vi u = \vi 0\).

In total, we obtained \((\vi v - c \vi v_1 \wdg \dots \wdg \vi v_n) \wdg \vi w = \vi 0\) for all \(\vi w \in \ExtProd^{n - k} V\), and by Lemma 3.1, it finally yields \(\vi v = c \vi v_1 \wdg \dots \wdg \vi v_n\).

Example 3.2 For illustration take \(V = \R^4\) and \(\vi v = \vi e_1 \wdg \vi e_2 + \vi e_3 \wdg \vi e_4\), which is notably not a product. Our goal will be to find \(\vi x = \sum_{i = 1}^4 x_i \vi e_i\) such that \(\vi v \wdg \vi x = \vi 0\). Then \[ \vi v \wdg \vi x = x_3 \vi e_1 \wdg \vi e_2 \wdg \vi e_3 + x_4 \vi e_1 \wdg \vi e_2 \wdg \vi e_4 + x_1 \vi e_1 \wdg \vi e_3 \wdg \vi e_4 + x_2 \vi e_2 \wdg \vi e_3 \wdg \vi e_4 = 0, \] and notice that each of the terms corresponds to a different basis vector in \(\ExtProd^3 \R^4\), i.e., they cannot subtract and thus \(\vi x = \vi 0\) (whereas if \(\vi v\) was a product, we would not have this restriction).

Let \(\vi A : V \to W\) be linear. Our goal will now be to extend this map to \(k\)-th exterior powers \(\ExtProd^k V\) and \(\ExtProd^k W\). To this end, we naturally define \[ \vi v_1 \wdg \dots \wdg \vi v_k \mapsto (\vi A \vi v_1) \wdg \dots \wdg (\vi A \vi v_k). \tag{3.7}\] Let now \(V\) be equipped with a scalar product, i.e., a non-degenerate bilinear form. This can also be extended to the exterior power \(\ExtProd^k V\). Indeed, take \(\vi v = \vi v_1 \wdg \dots \vi v_k \in \ExtProd^k V\) and \(\vi w = \vi w_1 \wdg \dots \wdg \vi w_k \in \ExtProd^k V\) and set \[ \scal {\vi v}{\vi w} \letDef \det \brackets{\scal{\vi v_i}{\vi w_j}_{i,j}}. \tag{3.8}\] Since the Gram matrix is regular for \(\vi v \neq \vi 0\), this scalar product is non-degenerate on \(\ExtProd^k V\), as we wanted.

Lastly, let \(V\) be an \(n\)-dimensional vector space with a scalar product and an orientation — By [1, prp. 15.3] orientation of a vector space is given by an ordered basis \(\vi v_1, \dots, \vi v_n\). Now every non-zero element \(\vi w \in \ExtProd^n (V^*)\) determines an orientation \(O_{\vi w}\) of \(V\) as follows:

• if \(n \geq 1\), then \(O_{\vi w}\) is the set of ordered bases \((\vi e_1, \dots, \vi e_n)\) such that \(O_{\vi w}(\vi e_1, \dots, \vi e_n) > 0\);
• if \(n = 0\), then \(O_{w}\) is \(+1\) if \(w > 0\), and \(-1\) if \(w < 0\).

Two nonzero \(n\)-covectors determine the same orientation if and only if each is positive multiple of each other! This, in turn, also implies that for 2 bases \(\vi v_1, \dots, \vi v_n\) and \(\vi w_1, \dots, \vi w_n\) of \(V\) we can write \(\vi v_1 \wdg \dots \wdg \vi v_n = c \vi w_1 \wdg \dots \wdg \vi w_n\) by Proposition 3.3 with \(c > 0\) if the bases are oriented the same.

Definition 3.5 (Hodge star functional) Every \(\vi \lmbd \in \ExtProd^k V\) defines a linear map \(L_{\vi \lmbd}: \ExtProd^{n-k} V \to \R\), where \[ \vi \mu \mapsto \vi \lmbd \wdg \vi \mu =: L_{\vi \lmbd}(\vi u) \underbrace{\vi e_1 \wdg \dots \wdg \vi e_n}_{\norm{\cdot} = 1}, \tag{3.9}\] where \(\vi e_1, \dots, \vi e_n\) is an oriented orthonormal basis of \(V\).

Since \(V\) has orthonormal basis, it also has a scalar product, and thus by (3.8) also \(\ExtProd^{n-k}\) is equipped with a non-degenerate scalar product². Hence, there exists \(\hodge \vi \lmbd \in \ExtProd^{n-k} V\) such that \(L_{\vi \lmbd}(\vi \mu) = \scal {\hodge \vi \lmbd} {\vi \mu}\) (by Theorem 4.5). This further induces an isomorphic linear map \(\hodge: \ExtProd^k V \to \ExtProd^{n-k} V\) called the Hodge star, i.e., \[ \vi \lmbd \wdg \vi \mu = \scal {\hodge \vi \lmbd} {\mu} \vi e_1 \wedges \vi e_n. \tag{3.10}\]

Important 3.1: Notation

Let us now introduce the following notation: For \(1 \leq j_1 < \dots < j_m \leq n\), we write \(H = \set{j_1, \dots, j_m}\) and \(\vi e_H = \vi e_{j_1} \wdg \dots \wdg \vi e_{j_m} \in \ExtProd^m V\), where \(\vi e_i\) denotes a basis vector of \(V\).

Let \(\vi e_i\) be an orthonormal basis (and we still require orientation). For \(K\) an index set, \(\countElements{K} = n-k\), disjoint to \(H\) and \(N = \oneToN{n}\) we have \(\vi e_H \wdg \vi e_K = \scal {\hodge \vi e_H} {\vi e_K} \vi e_N\) by Definition 3.5. Of course, if \(H\) and \(K\) are only disjoint if \(K = N \setminus H = \complementOf{H}\), otherwise we would get \(\vi e_H \wdg \vi e_K = 0\). In particular, as \(\vi e_i\) is an orthonormal basis, it yields (for \(K = \complementOf{H}\)) \[ \vi e_H \wdg \vi e_K = \sign(H,K) \vi e_N \implies \hodge \vi e_H = \sign(H,K) \vi e_K, \] where \(\sign(H,K)\) is the orientation of the basis with indices following the concatenation of \(H\) and \(K\).

Example 3.3 (Hodge star and cross product) Let us now restrict ourselves to \(\R^3\) for simplicity. Then \(\hodge \vi e_1 = \pm \vi e_2 \wdg \vi e_3\), which further implies \[ \vi e_1 \wdg \vi e_2 \wdg \vi e_3 = \scal{\pm \vi e_2 \wdg \vi e_3} {\vi e_2 \wdg \vi e_3} \vi e_1 \wdg \vi e_2 \wdg \vi e_3, \] hence \(\hodge \vi e_1 = + \vi e_2 \wdg \vi e_3\). Similarly, we can compute that \(\hodge \vi e_2 = - \vi e_1 \wdg \vi e_3\) and \(\hodge \vi e_3 = + \vi e_1 \wdg \vi e_2\). Moreover, analogous calculation immediately yields the following \[ \hodge (\vi e_1 \wdg \vi e_2) = + \vi e_3, \quad \hodge (\vi e_1 \wdg \vi e_3) = - \vi e_2, \quad \hodge (\vi e_2 \wdg \vi e_3) = + \vi e_1. \] The similarity with cross-product is not coincidental. In fact, \(\vi x \times \vi y = \hodge (\vi x \wdg \vi y)\) holds in \(\R^3\)!

In general, we have that \(\hodge \hodge \vi e_H = (-1)^{k(n-k)} \vi e_h\), and thus \(\hodge \hodge \hodge \hodge = \hodge^4 = \id\). Also, later³ we shall need the exterior power of the dual space \(V^*\). For \(l_1, \dots, l_k \in V^*\) is the exterior product \(l_1 \wdg \dots \wdg l_k\) a multilinear map \(V^k \to \R\) set by \[ l_1 \wdg \dots \wdg l_k (\vi v_1, \dots, \vi v_k) \letDef \det\brackets{(l_i(\vi v_j))_{i,j}}. \] This map is alternating, see Definition 3.3, in the input \((\vi v_1, \dots, \vi v_k)\), which justifies interpreting it as a linear functional on \(\ExtProd^k V\). In other words, any \(l_1 \wdg \dots \wdg l_k \in \ExtProd^k (V^*)\) is an element from the dual \((\ExtProd^k V)^*\), i.e., \(\ExtProd^k V^* \cong \ExtProd^k (V^*)\).

Important 3.2: Convention: Einstein summation

If one index occurs twice in an expression, once as an upper index and once as a lower index, this index is summed over the according index space. Lower indices are called covariant, whereas upper indices contravariant.

Example 3.4 Concretely, we may write \(\vi x = x^i \vi e_i = \sum_{i} x^i \vi e_i\). A bit more involved example may be for \(\countElements{H} = k\): \[ x^H \vi e_H = \sum_{\countElements{H} = k} x^H \vi e_H. \] Similarly, if \(\vi a = y_i \vi e^i = \sum_{i = 1}^n y_i \vi e^i\), where \(\vi e^i\) is the dual basis of \(V^*\), then \[ \vi a(\vi x) = x^i y_j \vi e^j(\vi e_i) = x^i y_j \dirac_i^j = x^j y_j. \]

Remark 3.7. Let \(V\) be a vector space with bases \((\vi e_i)_i\) and consider a vector \(\vi x \in V\), i.e., \(\vi x = x^i \vi e_i\). Now we shall define a new basis \((\vi e_j')_j\) by a linear change of coordinates \(\vi A = (a^i_j)_{i,j}\), thus \(\vi e_j' = a^i_j \vi e_i\). Notice that by invertibility of \(\vi A\) we get that \(\vi A\Inv = (a\Inv)^j_i\), i.e., \[ a^i_j \brackets{a\Inv}^j_k = \dirac^i_k. \]

We shall now write the vector \(\vi x\) in the basis \((\vi e_j')_j\) by its representation in \((\vi e_i)_i\). To this end, calculate \[ x^i \vi e_i = \vi x = x'^j \vi e_j' = x'^j a^i_j \vi e_i \implies x'^j = \brackets{a\Inv}^j_i x^i, \] hence the coordinates \(x^i\) transform with inverse of change-of-basis (which we call contravariant behavior). On the other hand, it can be analogously shown that coordinates of covectors transform the same as change-of-basis in the original vector space, i.e., the “co-vary” as the original basis making them covariant.

3.2 Differentiable Manifolds

Let \(\manifold\) be a topological space, which we shall equip with a certain structure. Consider \((U_i)_{i \in I}\) open sets in \(\manifold\) with the property \(\bigcup_{i \in I} U_i = \manifold\). Furthermore, we consider charts \((\vf_i, U_i)\), where \(\vf_i : U_i \to V_i \subseteq \R^n\) are homeomorphism⁴, such that they are compatible. Compatibility states that for \(U_i \cap U_j \neq \emptyset\) the mapping \[ \vf_i \after \vf_j\Inv : \vf_j(U_i \cap U_j) \to \vf_i(U_i \cap U_j) \in \Contf{\infty}, \tag{3.11}\] i.e., it is infinitely differentiable (for simplicity). Note that in theory, it is sufficient to require that \(\vf_i \after \vf_j\Inv\) is only a diffeomorphism⁵, however then we have to always “count” the number of allowed derivatives such that all the operations are well-defined. For illustration, see Figure 3.2.

Figure 3.2: Illustration of a differentiable manifold with an example of (a part of) an atlas

Let us call atlas the collection of all charts \((\vf_i, U_i)_{i \in I}\). Having an atlas allows us to define differentiability of functions \(f : \manifold \to \R\) by identifying it with the differentiability of \(f \after \vf_i\Inv : V_i \to \R\). Importantly, due to the compatibility of charts (3.11), the differentiability of \(f\) is well-defined on the intersections. Hence, an atlas defines a differentiable structure on \(\manifold\). We remark that the maximal atlas contains all possible chart maps and domains, and that it surely exists by Zorn’s lemma 4.2.

3.2.1 Tangent and Cotangent Spaces

So far, we have seen that an atlas provides a differentiable structure to \(\manifold\). However, it is not a priori trivially clear how to define tangents, as \(\manifold\) might not be a subset of \(\R^n\). Intuitively speaking, in this section we shall define the notion of said tangents by identifying directions with directional derivatives of functions on \(\manifold\).

Let us fix a point \(P \in \manifold\), for which have (multiple) \((u_0^1, \dots, u_0^n) \mapsto \vf_i\Inv (u_0^1, \dots, u_0^n) = P\) with \(\vf_i\Inv : V_i \to U_i\). Now consider \(f : \manifold \to \R\) and take \(f \after \vf_I\Inv : V_i \to \R\). This allows us to abuse the notation to write \(f(u_0^1, \dots, u_0^n)\) and we define the tangent space \(\tangentAt{P} \manifold\) at the point \(P\) as the vector space of all directional derivatives in \((u_0^1, \dots, u_0^n)\). In other words, we transform the problem of tangents of functions on the manifold \(\manifold\) to the tangents of functions on the chart domains \(V_i \subseteq \R^n\) (where we know how to do it).

Remark 3.8. An obvious basis for such space are \(\partialOp{u^i} = \partial_i\) at the point \(P\) — the span of which produces a vector space of operators. In particular, this gives us tangent vectors that can be applied functions. Note that while this basis depends on the chart we are using (by the \(u^i\)), the change of a chart corresponds only to a change of basis (by compatibility (3.11) are the charts isomorphic, even diffeomorphic, on the intersections). As such, fixing \(P\) and then choosing a corresponding chart is indeed a viable well-defined strategy.

Indeed, let \(P \in \manifold\) be such that \(P \in U_i \cap U_j\) with corresponding charts \((\vf_i, U_i)\) and \((\vf_j, U_j)\), recall Figure 3.2. Let \((u^1, \dots, u^n)\) be a basis of \(V_i\) and \((v^1, \dots, v^n)\) the basis of \(V_j\), then \[ \tangentAt{P} \manifold = \Span{\partialOp{u^1}, \dots, \partialOp{u^n}} = \Span{\partialOp{v^1}, \dots, \partialOp{v^n}} \quad \& \quad \partialOp{v^l} = \pDeriv{u^k}{v^l} \partialOp{u^k} \tag{3.12}\] at the point \(P\) (where we use the Einstein notation, see Important 3.2). This is obviously only a change of basis.

By collecting tangent spaces at all points of the manifold \(\manifold\) we construct the tangent bundle as \[ \tangent \manifold = \biguplus_{P \in \manifold} \tangentAt{P} \manifold, \] where \(\biguplus\) is the disjoint union, as the individual tangents spaces are not simply subsets of \(\R^n\), but are bound to their respective points \(P\) and they also carry this information. Finally, there exists a map \(\projection : \tangent \manifold \to \manifold\) such that \((P, \vi v) \mapsto P\) with \(\vi v \in \tangentAt{P} \manifold\).

Definition 3.6 (Vector field) A vector field \(\vi X\) is a section from the tangent bundle \(\tangent \manifold\), i.e., for each \(P \in \manifold\) we take a vector from \(\tangentAt{P} \manifold\). Thus \(\vi X = x^k \partialOp{u^k}\) in the corresponding chart \((\vf_i, U_i)\).

Clearly, the vector field \(\vi X\) is differentiable if coordinate functions \(x^k : V_i \to \R\) are differentiable in every chart (there we know what “differentiability” even means). Of course, the coordinate function changes by considering different charts (even for a given point, though keep in mind Remark 3.8).

For every function \(f : \manifold \to \R\) we can compute \(\vi X f\), i.e., every vector field acts on functions. In particular, we define the differential of \(f\) as \[ \dd f(\vi X) = \vi X f, \tag{3.13}\] which offers two different perspectives on the same object. Namely, the differential \(\dd f\) can be viewed as a linear functional on \(\tangentAt{P} \manifold\) for every \(P \in \manifold\) (recall that a vector space is only a section of the tangent bundle \(\tangent \manifold\)). In other words, this locally defines \(\dd f\) as an element of the dual space \(\tangentAt{P}^* \manifold\) by \[ \dd f = \pDeriv{f}{u^k} \dd u^k, \] where \(\dd u^k\) form a covector base of \(\tangentAt{P}^* \manifold\) (dual to the basis \(\partialOp{u^1}, \dots, \partialOp{u^n}\) of \(\tangentAt{P} \manifold\)). Moreover, for a basis covector \(\dd u^k\) holds \[ \dd u^k \brackets{\partialOp{u^l}} = \pDeriv{u^k}{u^l} = \dirac^k_l. \] As \(\dd u^k\) are covariant, they transform with the change-of-basis matrix (unlike the \(\partialOp{u^l}\) basis of contravariant vectors, see (3.12)), i.e., \(\dd u^k = \pDeriv{u^k}{v^l} \dd v^l\). Finally, we denote by \(\tangent^* \manifold = \biguplus_{P \in \manifold} \tangentAt{P}^* \manifold\) the cotangent bundle (as an analogue to the tangent bundle).

Tip

Recall that in Analysis 2, we had \[ f(u^1, \dots, u^n) = f(u_0^1, \dots, u_0^n) + \grad f(u_0^1, \dots, u_0^n) \mtr{u^1 - u_0^1 \\ \vdots \\ u^n - u_0^n} + \text{ error }. \] Here, we typically transport \(\dd f(u_0^1, \dots, u_0^n)\) via the scalar product to the tangent space (as the gradient \(\grad f(u_0^1, \dots, u_0^n)\)).

3.2.2 Differential Forms

First and foremost, one should note that expressions of form \(\omega = a_k \dd u^k\) (as such, also the basis covectors of \(\tangentAt{P}^* \manifold\) themselves) are differential forms of order \(1\) with \(\omega(P) \in \tangentAt{P}^* \manifold\). Moreover, every function \(f\) induces a differential form \(\omega = \d f\), and such forms are called exact forms — more precisely, a form \(\omega\) is called exact if it may be written as \(\omega = \d \alpha\), where \(\alpha\) is another differential form. However, the converse does not hold even with \(\Contf{\infty}\)! It is easy to see that when an exact form \(\omega = \omega_i \dd u^i\) corresponds to at least twice differentiable potential function \(f\)⁶, we obtain a nice necessary condition stemming from the symmetry of second derivatives of \(f\), \[\begin{gather*} \pDeriv{\omega_i}{u^j} = \partialOp{u^j} \pDeriv{f}{u^i} = \frac{\partial^2 f}{\partial u^i \partial u^j} = \partialOp{u^i} \pDeriv{f}{u^j} = \pDeriv{\omega_j}{u^i} \\ \; \\ \Updownarrow \end{gather*}\] \[ \pDeriv{\omega_i}{u^j} - \pDeriv{\omega_j}{u^i} = 0. \tag{3.14}\] A form \(\omega\) which follows this necessary condition (3.14) is called closed.

Remark 3.9 (Lie bracket). Let \(\vi X, \vi Y\) be vector fields and \(f\) be a function \(\manifold \to \R\), then \(\vi X f\), \(\vi Y f\) are also functions \(\manifold \to \R\). Thus \(\vi X \vi Y f\) and \(\vi Y \vi X f\) are again functions, but are \(\vi X \vi Y\) and/or \(\vi Y \vi X\) vector fields?

Take \(\vi X = x^k \partialOp{u^k}\) and \(\vi Y = y^k \partialOp{u^k}\), then \[ \vi X \vi Y f = x^k \partialOp{u^k} \brackets{ y^l \partialOp{u^l} f } = x^k \bigg( \underbrace{\pDeriv{y^l}{u^k} \pDeriv{f}{u^l}}_{\in \tangentAt{P} \manifold} + \underbrace{y^l \frac{\partial^2 f}{\partial u^l \partial u^k}}_{\notin \tangentAt{P} \manifold} \bigg), \] hence \(\vi X \vi Y\) is not a vector field (due to the inclusion of second order terms). However, one might consider the so called Lie bracket \(\lieBracket{\vi X}{\vi Y} = (\vi X \vi Y - \vi Y \vi X)\) of \(\vi X\) and \(\vi Y\), for which holds \[ \lieBracket{\vi X}{\vi Y} = \underbrace{\brackets{x^k \pDeriv{y^l}{u^k} \partialOp{u^l} - y^l \pDeriv{x^k}{u^l} \partialOp{u^k}}}_{\in \tangentAt{P} \manifold} f. \] Specifically, it is easy to see that the Lie bracket is symmetric.

Definition 3.7 (Differential form) A differential form of degree \(k\) is a section in \(\ExtProd^k \tangent^* \manifold = \biguplus_{P \in \manifold} \ExtProd^k \tangentAt{P}^* \manifold\).

A differential form may be expressed as \(\omega = a_H \dd u^H\) in every chart, where we use the notation introduced in Important 3.1. For illustration. for a degree \(k\) differential form the index set \(H = \set{h_1, \dots, h_k}\) goes through all \(k\)-element subsets of \(\oneToN{n}\), i.e., \(\countElements{H} = k\), and \(\dd u^H = \dd u^{h_1} \wdg \dots \wdg \dd u^{h_k}\). Concretely, \(\omega(P) \in \ExtProd^k \tangentAt{P}^* \manifold\) is a multilinear form on the tangent space, such that we can evaluate it as \(\omega(P)(\vi v_1, \dots, \vi v_k)\) for \(\vi v_1, \dots, \vi v_k \in \tangentAt{P} \manifold\).

Proposition 3.4 (Exterior derivative) There is exactly one linear map \(\d : \ExtProd^k \tangent^* \manifold \to \ExtProd^{k+1} \tangent^* \manifold\) with the following properties:

1. \(\d(\lmbd \wdg \mu) = \d \lmbd \wdg \mu + (-1)^{\degree \lmbd} \lmbd \wdg \d \mu\) (graded product/Leibniz rule);
2. \(\d (\d \omega) = 0\) (Poincaré lemma);
3. for \(f \in \ExtProd^0 \tangent^* \manifold\) (thus \(f\) is a function) \(\d(f) = \d f\) (coordinate-free by (3.13)⁷).

Let us note that the second property considered with functions \(f,g\) (i.e., 0-forms) reads as (recalling tensor product for functions) \[ \d (f g) = \d (f \wdg g) = g \dd f + f \dd g, \tag{3.15}\] hence we recover the standard product rule for differentiation.

Proof. We shall split the proof into 6 steps [2, thm. 30.4., 1, prp. 14.23. & thm 14.24.].

Step 1.: Let us verify uniqueness of \(\d\). Firstly, we show that properties 1. and 2. imply that for any forms \(\omega_1, \dots, \omega_k\), we have \(\d (\omega_1 \wedges \omega_k) = 0\). Let us proceed by induction over \(k\). The base case of \(k = 1\) is a consequence of 3. Now suppose it holds for \(k - 1\) and set \(\eta = (\d \omega_2 \wedges \d \omega_k)\), which by 1. can be re-written as \[ \d(\d \omega_1 \wdg \eta) = \d(\d \omega_1) \wdg \eta \pm \d \omega_1 \wdg \d \eta. \] The first term vanishes by 2., and the \(\d \eta = 0\) by the induction hypothesis. Furthermore, we show that for any \(k\)-form \(\omega\), the form \(\d \omega\) is determined entirely by the value of \(\d\) on 0-forms, which are, in turn, specified by 3. Since \(\d\) is linear, it suffices to consider the case \(\omega = f \d u^H\) (here \(f\) is a 0-form). Compute \[\begin{align*} \d \omega = \d (f \d u^H) &= \d (f \wdg \d u^H) \\ &\tOnTop{=}{1.} \d(f) \wdg \d u^H + f \wdg \d (\d u^H) \\ &\tOnTop{=}{3.} \d f \wdg \d u^H \end{align*}\] and the the result we have just proved (here the necessary form of 2.). This gives that \(\d \omega\) is determined solely by the value of \(\d\) on 0-form \(f\). This is a coordinate-free object, which we shall use to lift the restriction to a single chart (hence we can only study the object on a single chart domain).

Step 2.: We now define \(\d\). It’s value is specified by 3. Not only it is in accordance with the uniqueness results above, but it also gives us a recipe to define it for forms of positive degree: If \(V\) is open set in \(\R^n\) and if \(\omega\) is a \(k\)-form on \(V\) (i.e., on a given chart domain), we write \(\omega\) uniquely in the form \(\omega = f_H \d u^H\), and define \[ \d \omega = \d \brackets{f_H \d u^H} \letDef \d f_H \wdg \d u^H. \tag{3.16}\] Let us note that here \(\d u^H\) come from the basis representation, and thus all \(H\)’s are ascending sets of indices. We check that \(\d \omega\) is indeed infinitely differentiable. For this purpose, we first compute \[ \d \omega = \brackets{\pDeriv{f_H}{u^i} \d u^i} \wdg \d u^H, \] where terms with \(i \in H\) will vanish. The \((k+1)\)-forms of remaining terms can be rearranged so that the indices are in ascending order (producing only \(\pm 1\)). In total, we see that it is a linear combination of \((k+1)\)-forms with coefficients \(\pDeriv{f_H}{u^i}\). Then smoothness of \(f\) implies \(\d \omega\) is itself infinitely differentiable.

Now let us show \(\d\) is linear on \(k\)-forms with \(k > 0\). Let \[ \omega = f_H \d u^H \quad \& \quad \eta = g_H \d u^H \] be \(k\)-forms, then by (3.16) and the fact that such \(\d\) is linear on 0-forms \[\begin{align*} \dd (a \omega + b \eta) &= \dd \brackets{\brackets{a f_H + b g_H} \d u^H} \\ &= \dd \brackets{a f_H + b g_H} \wdg \d u^H \\ &\tOnTop{=}{linear} (a \dd f_H + b \dd g_H) \wdg \d u^H \\ &= a \dd \omega + b \dd \eta. \end{align*}\]

Step 3.: We now show that if \(J\) is an arbitrary \(k\)-tuple of indices from \(\oneToN{n}\), then \(\d (f \wdg \d u^J) = \d f \wdg \d u^J\). Certainly, this holds if two of the indices in \(J\) are the same, since \(\d u^J = 0\) in such case. Thus suppose the indices in \(J\) are distinct and let \(H\) be the index set obtained by rearranging indices of \(J\) in ascending order (by permutation \(\pi \in S_k\)). As the exterior product \(\wedge\) is alternating, see Definition 3.3 and Definition 3.4, it implies \(\d u^H = \sign(\pi) \d u^J\). The definition (3.16) of \(\d\) in Step 2. together with its linearity now yield \[ \sign(\pi) \d (f \wdg \d u^J) = \sign(\pi) \d f \wedge \d u^J. \]

Step 4.: Let us focus on the graded product rule property 1. In the case both \(\mu, \lmbd\) are 0-forms, the graded product rule simplifies to the standard product rule, which we have shown as a small remark in (3.15). Consider \(\mu, \lmbd\) be \(k\)-forms of positive order (i.e., both are non-zero-forms) and by linearity of \(\d\), it suffices to consider the case \(\lmbd = f \d u^I\) and \(\mu = g \d u^J\). Then \[\begin{align*} \d (\lmbd \wdg \mu) &= \d \brackets{fg \d u^I \wdg \d u^J} \\ &\tOnTop{=}{Step 3.} \d (fg) \wdg \d u^I \wdg \d u^J \\ &= (g \dd f + f \dd g) \wdg \d u^I \wdg \d u^J \\ &= (\d f \wdg \d x^I) \wdg (g \wdg \d x^J) + (-1)^{k} (f \wdg \d u^I) \wdg (\d g \wdg \d u^J) \\ &= \d \lmbd \wdg \mu + (-1)^k \lmbd \wdg \d \mu, \end{align*}\] where the \((-1)^k\) factor comes from the fact that we need to exchange the \(k\)-form \(\d u^I\) and the 1-form \(\d g\). Finally, the case where one of the forms is actually a function (i.e., a 0-form) proceeds analogously to the case we have just shown.

Step 5.: Let us finally focus on the Poincaré’s lemma property 2. Assume, for now, that \(f\) is a differential 0-form, then \[\begin{align*} \d (\d f) &\fromDef \d \brackets{\pDeriv{f}{u^i} \d u^i} \\ &\tOnTop{=}{Step 2.} \d \brackets{\pDeriv{f}{u^i}} \d u^i \\ &= \brackets{\partialOp{u^j}\partialOp{u^i} f} \d u^j \wdg \d u^i, \end{align*}\] and since by Schwarz’ theorem the second derivatives of a smooth function \(f\) are symmetric, but the exterior product is alternating (or anti-symmetric), we get the desired result. Note that we can also rewrite this as \[ \d(\d f) = \sum_{i < j} \brackets{ \frac{\partial^2}{\partial u^i \partial u^j} - \frac{\partial^2}{\partial u^j \partial u^i} } f \d u^i \wdg \d u^j. \]

Step 6.: Last, but not least, we show that if \(\omega\) is a differential \(k\)-form with \(k > 0\), we still get \(\d (\d \omega) = 0\). Since \(\d\) is linear, it suffices to focus on \(\omega = f \d u^H\). Then (3.16) implies \[ \d(\d \omega) = \d \brackets{\d f \wdg \d u^H} \tOnTop{=}{1.} \d(\d f) \wdg \d u^H - \d f \wdg \d (\d u^H), \] where \(\d(\d f) = 0\) by Step 5., and \[ \d (\d u^H) = \d (1 \d u^H) = \d(1) \wdg \d u^H = 0 \] by definition. Hence \(\d(\d \omega) = 0\) and this concludes the proof.

Example 3.5 Let us turn our attention to the problem of integrating a vector field (e.g., a line integral or a surface integral). Let \((P, Q, R)\) be the coordinates of a given vector field in \(\R^3\), and thus \(\d x, \d y, \d z\) be the coordinates of motion, we then have \[ \int P \d x + Q \d y + R \d z, \] where the integrand takes form a scalar product, and is obviously a differential form \(\omega\). then \[\begin{align*} \d \omega &= \brackets{\pDeriv{P}{x} \d x + \pDeriv{P}{y} \d y + \pDeriv{P}{z} \d z} \wdg \d x + \brackets{\pDeriv{Q}{x} \d x + \pDeriv{Q}{y} \d y + \pDeriv{Q}{z} \d z} \wdg \d y + \brackets{\pDeriv{R}{x} \d x + \pDeriv{R}{y} \d y + \pDeriv{R}{z} \d z} \wdg \d z \\ &= \brackets{\pDeriv{R}{y} - \pDeriv{Q}{z}} \d y \wdg \d z + \brackets{\pDeriv{P}{z} - \pDeriv{R}{x}} \d z \wdg \d x + \brackets{\pDeriv{Q}{x} - \pDeriv{P}{y}} \d x \wdg \d y, \end{align*}\] where we get the coordinates of the curl of the given vector field.

Similarly for a \(2\)-form, \[\begin{align*} \mu &= P \d y \wdg \d z + Q \d z \wdg \d x + R \d x \wdg \d y \\ \d \mu &= \pDeriv{P}{x} \d x \wdg \d y \wdg d z + \pDeriv{Q}{y} \d y \wdg \d z \wdg \d x + \pDeriv{R}{z} \d z \wdg \d x \wdg \d y \\ &= \brackets{\pDeriv{P}{x} + \pDeriv Q y + \pDeriv R z} \d x \wdg \d y \wdg \d z, \end{align*}\] we get the divergence of the vector field instead!

We shall now study the relationship between the exact and closed forms, in particular we shall show that locally every closed form is exact!

Let \(\kformsOver{k}{X}\) denote the space of smooth differential forms of degree \(k\) on \(X\). Moreover, we shall define a “pullback”, firstly for smooth functions, and then also for norms.

Definition 3.8 (Pullback of functions) Given a smooth map \(f : N \to M\) between differentiable manifolds \(M, N\) and a smooth function \(\vf : M \to \R\), the pullback \(\pullback{f} \vf\) is the function defined by \((\pullback{f} \vf)(\vi x) = \vf(f(\vi x))\) for \(x \in N\).

Definition 3.9 (Pullback of differential forms) Let \(f : N \to M\) be a smooth map between differentiable manifolds \(M, N\) and let \(\omega \in \kformsOver{k}{M}\) be a smooth \(k\)-form on \(M\). The pullback \(\pullback{f} \omega \in \kformsOver{k}{N}\) is the unique differential \(k\)-form on \(N\) such that (for a given point \(P \in N\)) \[ (\pullback f \omega)_P(\vi v_1, \dots, \vi v_k) = \omega_{f(P)}(\d f_P(\vi v_1), \dots, \d f_P(\vi v_k)), \] where \(\vi v_i \in \tangentAt{P} M\). Here, \(\d f_P : \tangentAt{P} N \to \tangentAt{f(P)} M\) is the differential pushforward of \(f\) at \(P\).

From linearity of the \(k\)-form follows also linearity of the pullback. Lastly, let \(\oper K : \kformsOver{p+1}{I \times U} \to \kformsOver{p}{U}\), where \(I = [0, 1]\), be the homotopy operator, which maps \[ \begin{aligned} A \d u^H &\mapsto 0 \\ B(t, \vi x) \dd t \wdg \d u^J &\mapsto \int_0^1 B(t, \vi u) \dd t \dd u^J, \end{aligned} \tag{3.17}\] where \(B \in \Contf{\infty}\).

Proposition 3.5 If one recalls \(j_0 : U \to I \times U\) with \(\vi u \mapsto (0, \vi u)\) and analogously \(\vi u \onTop{\mapsto}{j_1} (1, \vi u)\), it then follows \(\oper K(\d \omega) + \d \oper K(\omega) = \pullback{j_1} \omega - \pullback{j_0} \omega\).

Proof. Firstly, let us note that by a simple calculation \[ \pullback{j_1}(\d u^i) = \d u^i, \quad \pullback{j_1}(\d t) = 0, \quad \pullback{j_0}(\d u^i) = \d u^i, \quad \pullback{j_0}(\d t) = 0. \] Consider now the simpler case of \(\omega = A \d u^H\), where, by definition of \(\oper K\), we have that \(\omega\) maps to \(\oper K\omega = 0\) and \[ \d \omega \fromDef \d A \wedge \d u^H = \pDeriv{A}{t} \d t \wdg \d u^H + \text{ other terms without } \d t, \] thus \(\oper K(\d \omega) = \int_0^1 \pDeriv{A}{t} \dd t \dd u^H = \brackets{A(1, \vi u) - A(0, \vi u)} \dd u^H\) as the rest of the terms without \(\d t\) is annihilated by \(\oper K\). In total, \(\oper K(\d \omega) + \d \oper K(\omega) = \brackets{A(1, \vi u) - A(0, \vi u)} \dd u^H = \pullback{j_1} \omega - \pullback{j_0}\omega\).

Turning our attention to \(\omega = B \dd t \wdg \d u^J\) firstly yields (notice that the dependence of \(B\) on \(t\) always vanishes by \(\wdg \d t\)) \[ \d \omega = \d B \wdg \d t \wdg \d u^J = \pDeriv{B}{u^i} \d u^i \wdg \d t \wdg \d u^J = - \pDeriv{B}{u^i} \d t \wdg \d u^i \wdg \d u^J, \] hence \(\oper K(\d \omega) = - \int_0^1 \pDeriv{B}{u^i}(t, \vi u) \dd t \dd u^i \wdg \d u^J\). Additionally, \[ \oper K \omega \fromDef \int_0^1 B(t, \vi u) \dd t \dd u^J \implies \d (\oper K \omega) = \int_0^1 \pDeriv{B}{u^i} (t, \vi u) \dd t \dd u^i \wdg \d u^J, \] which, combined together with the last result, produces \(\oper K(\d \omega) + \d (\oper K \omega) = 0\). This is indeed consistent, as for \(\omega = B \d t \wdg \d u^J\) also \[ \pullback{j_i} \omega \fromDef \underbrace{\pullback{j_i} (\d t)}_0 \wdg \pullback{j_i} (B \d u^J) = 0. \]

Definition 3.10 (Contractible set) A subset \(U \subseteq \R^n\) is called contractible if there exists a continuous map, called null-homotopy, \(\phi : I \times U \to U\) with properties

1. \(\phi(1, \vi x) = \vi x\) for all \(\vi x \in U\);
2. \(\phi(0, \vi x) = \vi x_0\) for all \(\vi x \in U\) and some fixed \(\vi x_0 \in U\).

If \(U\) is contractible, we can make the following computation: \[ (\phi \after j_1)(\vi x) = \vi x \quad \& \quad (\phi \after j_0)(\vi x) = \vi x_0. \] Let now \(\omega\) be a \(k\)-form on \(U\), then \[ \underbrace{(\pullback{j_1} \after \pullback{\phi})}_{\pullback{(\phi \after j_1)}} \omega = \omega \quad \& \quad (\pullback{j_0} \after \pullback{\phi}) \omega = 0. \] We can apply Proposition 3.5 to \(\pullback{\phi} \omega\), where we now consider only closed forms (i.e., \(\d \omega = 0\)), which tells us \[ \oper K (\underbrace{\d \pullback{\phi} \omega}_{\fromDef \pullback{\phi} \d \omega}) + \d \oper K(\pullback{\phi} \omega) = \pullback{j_1} \pullback{\phi} \omega - \pullback{j_0} \pullback{\phi} \omega = \omega, \] and by \(\d \omega = 0\), it finally yields \(\omega = \d (\oper K \pullback{\phi} \omega)\), i.e., \(\omega\) is exact! In other words, for contractible domains the notions of exact and closed differential forms coincide! Note that we have shown that every exact form is also closed earlier.

Example 3.6 When solving differential equations, e.g., in Analysis 2, one may sometimes use the mapping \(t \mapsto t (\xi, \eta)\) to compute the line integral of \(\omega = P \d \xi + Q \d \eta\), \[ \int_{(0,0)}^{(\xi, \eta)} P \d \xi + Q \d \eta = \int_0^1 \brackets{P(t\xi, t \eta) \xi + Q(t \xi, t \eta) \eta} \dd t, \] which was often much easier to compute. Note that this works precisely because a line (through origin) is contractible to the origin!

3.3 Integration of Differential Forms

So far, we introduced the probleme where one has a vector field, and wants to compute its integral over a surface (there we need a normal of the surface) or over a curve (there we need a tangent to the curve) — notice, however, that in both times orientation is needed!

Definition 3.11 (Oriented manifold) An \(n\)-dimensional manifold \(\manifold\) is oriented, if there exists an \(n\)-form \(\omega\) on \(\manifold\), such that \(\omega(P) \neq 0\) for every \(P \in \manifold\).

Remark 3.10. At every \(P \in \manifold\) we can use \(\omega\) to check orientation of the basis \((\vi v_1, \dots, \vi v_n)\) of \(\tangentAt{p} \manifold\) by checking the sign of \(\omega(P)(\vi v_1, \dots, \vi v_n).\)

Remark 3.11 (Oriented atlas). Let \(\atlas = (U_i, \vf_i)_{i \in I}\) be an atlas. If for every \(i,j\) with \(U_i \cap U_j \neq \emptyset\) it holds that⁸ \(\det \brackets{\frac {\partial (\vf_i \after \vf_j\Inv)} {\partial(u^1, \dots, u^n)}} > 0\), we call \(\atlas\) oriented. Moreover, manifold \(\manifold\) is oriented⁹ if it admits an oriented atlas.

Let \(\vi v_1, \dots, \vi v_n\) be an oriented basis of \(\R^n\) and consider the oriented \(n\)-simplex \(S\), defined as¹⁰ \[ S = \set{t^1 \vi v_1 + \dots + t^n \vi v_n \divider t^i \geq 0, \; \sum_{i = 1}^n t^i = 1}. \tag{3.18}\] Note that the orientation of the simplex \(S\) comes from the orientation of the basis \(\vi v_1, \dots, \vi v_n\). Then for an \(n\)-form \(\omega = A(\vi x) \d x^1 \wedges \d x^n\) we set \[ \int_{S} \omega \letDef \idotsint_S A(\vi x) \dd x^1 \dots \dd x^n, \tag{3.19}\] where the right-hand side is a Riemann integral! Importantly, notice that using this definition, we can integrate \(n\)-forms over \(n\)-simplices.

Now we shall study how integrals of differential forms on \(\manifold\) over (general) simplices behave, and how we could generalize this. To this end, consider \(m\)-simplex \(S\) in \(\R^m\) with \(m \leq n\), which can embedded into \(\R^n\) by embedding \(\iota: (t^1, \dots, t^m) \mapsto (t^1, \dots, t^m, 0, \dots, 0) \in \R^n\). Further, denote by \(T\) the image of \(S\) under \(\vf_i\Inv \after \iota\) (i.e., the image of \(m\)-dimensional simplex on \(n\)-dimensional manifold \(\manifold\) by \(\vf_i\Inv\)). Lastly, take \(\omega\) and \(m\)-form on \(\manifold\) given by \(\omega = a_H \d u^H\) in \(U_i\). By pullback of \(\iota\), i.e., \(\pullback{\iota} \omega\) we get an \(m\)-form on \(\R^m\), hence analogously to (3.19) we write \(\int_T \omega \letDef \int_S \pullback{\iota} \omega\). Figure 3.3 illustrates this setting visually.

Figure 3.3: Illustration of an \(m\)-simplex embedded into \(\R^n\) and further projected to \(T\) on the manifold \(\manifold\)

Remark 3.12. Using the construction above, we can integrate over diffeomorphic images of simplices.

Definition 3.12 (Chain) A chain \(C\) is a formal sum of (images of) simplices, \(C = \sum_{i} \alpha_i S_i\), where \(S_i\) are simplices and \(\alpha_i \in \R\) (or \(\Z\)).

Denoting \(\phi = \vf_i\Inv \after \iota\), we can extend integration over simplices to integration over chains as follows, \[ \phi(C) = \sum_{i} \alpha_i \phi(S_i) \implies \int_c \pullback{\phi}(\omega) = \sum_{i} \alpha_i \int_{S_i} \pullback{\phi}(\omega) \fromDef \sum_{i} \alpha_i \int_{\phi(S_i)} \omega =: \int_{\phi(C)} \omega. \]

Say we have a simplex \(S = \set{\sum_{i = 0}^n t^i P_i \divider t^i \geq 0, \; \sum_{i = 0}^n t^i = 1}\) in \(\R^n\), which has \(n+1\) indices \((P_0, \dots, P_n)\) and their order determines the orientation of the simplex. In other words, for \(\pi \in S_{n+1}\) we have \((P_{\pi(0)}, \dots, P_{\pi(n)}) = \sign(\pi) (P_0, \dots, P_n)\). Before rigorously defining the boundary \(\boundaryOf S\) of a simplex \(S\), we first show it on simple examples, see Figure 3.4:

1. \(n = 1\) (Figure 3.4 a)): \(\boundaryOf S: (P_1) - (P_0)\), where \((P_1), (P_0)\) are 0-dimensional simplices;
2. \(n = 2\) (Figure 3.4 b)): \[ \boundaryOf S: (P_0, P_1) + (P_1, P_2) + (P_2, P_0) = (P_1, P_2) - (P_0, P_2) + (P_0, P_1), \] where after rearranging the \(k\)-th term is missing exactly the vertex \(P_k\);
3. \(n = 3\) (Figure 3.4 c)): \[\begin{align*} \boundaryOf S: & (P_1, P_2, P_3) + (P_0, P_1, P_3) + (P_2, P_0, P_3) + (P_0, P_2, P_1) \\ &= (P_1, P_2, P_3) - (P_0, P_2, P_3) + (P_0, P_1, P_3) - (P_0, P_1, P_2). \end{align*}\]

Figure 3.4: Visualizations of basic simplices for \(n \in \oneToN{3}\)

Hence, in general for a simplex \(S = (P_0, \dots, P_n)\) we have \[ \boundaryOf S \letDef \sum_{i = 0}^n (-1)^i (P_0, \dots, P_{i - 1}, P_{i+1}, \dots, P_n) = \sum_{i = 0}^n (-1)^i (P_0, \dots, \cancel{P_i}, \dots, P_n). \tag{3.20}\] Furthermore, let us observe that \(\boundaryOf \boundaryOf S = 0\); indeed, compute \[ \begin{aligned} \boundaryOf \boundaryOf S &\fromDef \sum_{i = 0}^n (-1)^i \boundaryOf (P_0, \dots, \cancel{P_i}, \dots, P_n) \\ &= \sum_{i = 0}^n (-1)^i \brackets{\sum_{j = 0}^{i - 1} (-1)^j (P_0, \dots, \cancel{P_j}, \dots, \cancel{P_i}, \dots, P_n) - \sum_{j = i+1}^n (-1)^j (P_0, \dots, \cancel{P_i}, \dots, \cancel{P_j}, \dots, P_n)} \\ &= 0, \end{aligned} \tag{3.21}\] as we iterate through all combinations of \(i,j\), which subtract to result in \(0\). Moreover, this extends to chains right away by linearity, i.e., \(\boundaryOf \boundaryOf C = 0\).

Definition 3.13 (Cycle & boundary) Let \(C\) be a \(p\)-chain¹¹, then

• \(C\) is called a cycle, if \(\boundaryOf C = 0\);
• \(C\) is called a boundary, if there is a \((p+1)\)-chain \(D\) such that \(\boundaryOf D = C\).

Lemma 3.2 Every boundary is a cycle (but not vice versa).

Proof. Follows trivially from the chain version of (3.21).

Theorem 3.1 (Stokes’) Let \(C\) be a \(p\)-chain and \(\omega \in \kformsOver{p-1}{\manifold}\), then \(\int_C \dd \omega = \int_{\boundaryOf C} \omega\).

Example 3.7 Before proving the Stokes’ theorem 3.1 in full generality, we shall present several special cases, where it reduces to another theorem from Analysis 2 or 3.

Right away, for \(n = 2\) and \(p = 2\) we get (here, \(C\) is surface in \(\R^2\) and \(\boundaryOf C\) is a line in \(\R^2\), thus a curve in \(\manifold\)) \[ \int_{\boundaryOf C} P \dd x + Q \dd y = \int_C \brackets{\pDeriv{P}{x} \dd x + \pDeriv{P}{y} \dd y} \wdg \d x + \brackets{\pDeriv{Q}{x} -\pDeriv{P}{y}} \dd x \wdg \d y, \] which, one might recall, is the Gauss’ theorem.

Furthermore, for \(n = 3\) and \(p=2\) it yields \[\begin{align*} \int_{\boundaryOf C} P \dd x + Q \dd y + R \dd z &= \int_C \brackets{\pDeriv{P}{x} \dd x + \pDeriv{P}{y} \dd y + \pDeriv{P}{z} \dd z} \wdg \d x + \dots \\ &= \int_C \brackets{\pDeriv{R}{y} - \pDeriv{Q}{z}} \dd y \wdg \d z + \brackets{\pDeriv{P}{y} - \pDeriv{R}{x}} \dd z \wdg \d x + \brackets{\pDeriv{Q}{x} - \pDeriv{P}{y}} \dd x \wdg \d x, \end{align*}\] which the Stokes’ theorem as we know it from introductory analysis lectures.

If we take \(n = 3\) and \(n = 3\) \[ \int_{\boundaryOf C} P \dd y \wdg \d z + Q \dd z \wdg \d x + R \dd x \wdg \d y = \brackets{\pDeriv{P}{x} + \pDeriv{Q}{y} + \pDeriv{R}{z}} \dd x \wdg \d y \wdg \d z, \] we get the so called divergence theorem.

Lastly, for \(n = 1\), thus necessarily \(p = 1\), the form \(\omega\) now becomes of degree \(0\), i.e., a function \(f\), hence for \(C = (P_0, P_1)\) we get \[ \int_C \dd f = f(P_1) - f(P_0). \] In other words, we have generalized the fundamental theorem of calculus.

Proof. By linearity, it suffices to prove the theorem for a \(p\)-form \(\omega = A \d x^1 \wedges \d x^p\), in \(\R^{p+1}\), and \(C = (P_0, \dots, P_{p+1})\) (hence boundary is \(p\)-dimensional) with \(P_0\) and \(P_j = \big(0, \dots, \underbrace{1}_j, \dots, 0\big)\Tr\). Then \(\d \omega = (-1)^p \pDeriv{A}{x^{p+1}} \d x^1 \wedges \d x^{p+1}\) and \[ \boundaryOf C = \sum_{i = 0}^{p+1} (-1)^i (P_0, \dots, \cancel{P_i}, \dots, P_{p+1}) = (P_1, \dots, P_{p+1}) + (-1)^{p+1} (P_0, \dots, P_{p}) + \text{ other summands }. \tag{3.22}\] Firstly notice that \(\int_S \omega = 0\) if \(S\) if one of the “other summands”, as one of the variables, say \(x^j\), is constant \(0\) for some \(j = 1, \dots, p\), and as such \(\d x^j = 0\). Recall the definition of \(S\) (3.18), where the conditions \(x^j \geq 0 \land \sum_{j = 1}^{p+1} x^j = 1\) are equivalent to \(0 \leq x^{p+1} \leq 1- \sum_{j = 1}^p x^j\) (such that the sum is at most 1). Hence, by (3.19) \[ \begin{aligned} \int_C \d \omega &= (-1)^p \int_C \pDeriv{A}{x^{p+1}} \dd x^1 \dots \d x^{p+1} \\ &= (-1)^p \int_{\sum_{j = 1}^p x^j \leq 1} \int_{x^{p+1} = 0}^{1 - \sum_{j = 1}^p x^j} \pDeriv{A}{x^{p+1}} \dd x^{p+1} \dd x^1 \dots \d x^p \\ &\tOnTop{=}{Fubini} \int_{\sum_{j = 1}^p x^j \leq 1} \brackets{A\brackets{x^1, \dots, x^p, 1 - \sum_{j = 1}^p x^j} - A\brackets{x^1, \dots, x^p, 0}} \d x^1 \dots \d x^p. \end{aligned} \tag{3.23}\] Moreover, simplex definition (3.18) yields \[\begin{align*} (P_1, \dots, P_{p+1}) &= \set{(x^1, \dots, x^{p+1}) \divider x^1 + \dots + x^{p+1} = 1, \; x^j \geq 0}, \\ (P_0, \dots, P_{p}) &= \set{(x^1, \dots, x^{p}, 0) \divider x^1 + \dots + x^{p} \leq 1, \; x^j \geq 0}, \end{align*}\] as for \((P_0, \dots, P_{p})\) we have \(\sum_{j = 0}^p x^j = 1\). Thus, by (3.19) we also get \[ \begin{aligned} (-1)^{p+1} \int_{(P_0, \dots, P_p)} \omega &= (-1)^{p+1} \int_{x^1 + \dots + x^p \leq 1} A(x^1, \dots, x^p, 0) \dd x^1 \dots x^p \\ &\& \\ \int_{(P_1, \dots, P_{p+1})} \omega &= \int_{(P_1, \dots, P_0)} A\brackets{x^1, \dots, x^p, 1 - \sum_{j = 1}^p x^j} \dd x^1 \dots \d x^p \\ &= (-1)^p \int_{(P_0, \dots, P_p)} \dd x^1 \dots \d x^p, \end{aligned} \tag{3.24}\] where for the second integral we first fixed \(x^{p+1}\) by \(x^1, \dots, x^p\) in \(A\), which then allowed us to project the simplex \((P_1, \dots, P_{p+1})\) to \((P_1, \dots, P_p, P_0)\).

Finally, comparing (3.23) with (3.24) plugged into (3.22) produces the desired equality.

Let us remark that for each step of the proof of Stokes’ theorem 3.1 we need the fundamental theorem of calculus.

Remark 3.13. In particular, the integral gives a pairing between \(p\)-chain \(C\) and \(p\)-form \(\omega\), namely \((C, \omega) \mapsto \int_C \omega\). Moreover, taking \(C = \tilde{C} + \boundaryOf D\) and \(\omega = \tilde{\omega} + \d \eta\) where \(\tilde{C}\) is a cycle 3.13 and \(\tilde{\omega}\) is closed, yields \[ \int_C \omega = \int_{\tilde{C} + \boundaryOf D} \tilde{\omega} + \d \eta = \int_{\tilde{C}} \tilde{\omega} + \int_{\boundaryOf D} \tilde{\omega} + \int_{\tilde{C}} \d \eta + \int_{\boundaryOf{D}} \d \eta \implies \int_C \omega = \int_{\tilde{C}} \tilde{\omega}, \] as by Theorem 3.1 we have

• \(\tilde{C}\) is a cycle, i.e., \(\boundaryOf \tilde{C} = 0\) \(\implies\) \(\int_{\tilde{C}} \d \eta = \int_{\boundaryOf \tilde{C}} \eta = 0\);
• \(\tilde{\omega}\) is closed, then \(\int_{\boundaryOf D} \tilde{\omega} = \int_{D} \d \tilde{\omega} = 0\);
• \(\int_{\boundaryOf D} \d \eta = \int_{\boundaryOf \boundaryOf D} \eta = 0\) by (3.21).

In other words, this pairing of \(p\)-cycles and \(p\)-forms is isomorphic to an analogous pairing of quotient groups of cycles modulo boundaries and forms modulo closed forms.

Remark 3.14. So far, we have seen that while every exact form is closed, the converse is not true in general. However, locally every closed form is indeed exact (Poincaré’s lemma, see Proposition 3.5 and Definition 3.10), which we have shown for contractible neighborhoods of points. However, a stronger statement may be formulated using cycles!

Definition 3.14 (Period of a form) Let \(\omega\) be a \(k\)-form on \(\manifold\) and \(C\) be a \(k\)-cycle on \(\manifold\). Then the integral \(\int_C \omega\) is called a period (of \(\omega\)).

Theorem 3.2 (de Rham) A closed form is exact if and only if all its periods vanish.

As a special case follows the path independence \(\int_{\curve} \vi F \dd \vi r\) of a conservative vector field \(\vi F\) — indeed, for us this translates into integrals over \(1\)-forms, and by Theorem 3.2 this property follows.

3.4 Riemannian Metric on \(\mathcal{M}\)

Let us now equip every tangent space \(\tangentAt{P} \manifold\) with a norm (i.e., a non-degenerate bilinear form), which might be achieved by prescribing scalar products \(\scal{\partialOp{u^i}} {\partialOp{u^j}} =: g_{ij} = g\brackets{\partialOp{u^i}, \partialOp{u^j}}\) of basis vectors of \(\tangentAt{P} \manifold\). Here \(g_{ij}\) are local to the point \(P\) of \(\tangentAt{P} \manifold\) and we assume \((g_{ij})_{ij}\) is strictly positive definite. Note that \(g_{ij}\) are functions of the point \(P\) (given a suitable coordinate chart), and we require them to be infinitely differentiable.

A scalar product on the tangent space \(\tangentAt{P} \manifold\) canonically induces a corresponding scalar product on the cotangent space \(\tangentAt{P}^* \manifold\), i.e., for \(g_{ij} = g\brackets{\partialOp{u^i}, \partialOp{u^j}}\) we have in the dual basis \(g(\d u^i, \d u^j) =: g^{ij}\) with \(g^{il} g_{lj} = \dirac_j^i\). In other words, the matrices \((g^{ij})_{ij}\) and \((g_{ij})_{ij}\) are inverse. Additionally, this scalar product also induces a scalar product on \(\ExtProd^k \tangentAt{P}^* \manifold\) by ground determinants \[ g(\d u^H, \d u^K) = \det \brackets{(g^{ij})_{i \in H, j \in K}}, \] where \(H, K\) are subsets of \(\oneToN{n}\) of cardinality \(k\).

By equipping \(\ExtProd^k \tangentAt{P}^* \manifold\) with a scalar product, thus a norm and an orientation, we can define the Hodge-star operator 3.5, see (3.10). Recall that for a vector space \(V\) and \(\vi \alpha \in \ExtProd^k V\), \(\vi \beta \in \ExtProd^{n-k} V\), the Hodge-star operator \(\hodge : \ExtProd^k V \to \ExtProd^{n-k} V\) it holds \[ (\hodge \vi \alpha) \wdg (\hodge \vi \beta) \fromDef \scal{\hodge \hodge \vi \alpha} {\hodge \vi \beta} \vi \omega = (-1)^{k(n-k)} \scal {\vi \alpha} {\hodge \vi \beta} \vi \omega = (-1)^{k(n-k)} \vi \beta \wdg \vi \alpha = \vi \alpha \wdg \vi \beta, \] where the last equality holds as \(\wdg\) is alternating. Combining \(\vi \alpha \wdg \vi \beta \fromDef \scal{\hodge \vi \alpha} {\vi \beta} \vi \omega\) with an intermediate result of the above computation yields that for \(\hodge : \ExtProd^k V \to \ExtProd^{n-k} V\) the “complementary” Hodge-star operator \(\hodge : \ExtProd^{n-k} V \to \ExtProd^k V\) is also the adjoint.

Let us use these results to understand the surface of the manifold. For this purpose, we define a surface measure on an oriented manifold \(\manifold\), i.e., we have \(\tilde{\omega} \in \kformsOver{n}{\manifold}\) such that \(\tilde{\omega}(P) \neq 0\) for all \(P \in \manifold\). We define the volume form¹² by \(\d \vol = \omega = \frac {\tilde{\omega}}{\norm{\tilde{\omega}}}\) where \(\norm{\tilde{\omega}} = \sqrt{g(\tilde{\omega}, \tilde{\omega})}\). Specifically, the form \(\omega\) assigns volume 1 to a unit cube. Furthermore, we would like to write as multiple of \(\d u^1 \wedges \d u^n\). To this end, notice \(\norm{\d u^1 \wedges \d u^n}^2 = \det ((g^{ij})_{ij}) = \det ((g_{ij})_{ij})\Inv\). Thus indeed \(\omega = \sqrt{\det((g_{ij})_{ij})} \d u^1 \wedges \d u^n\).

We use the volume form to define a “pre-Hilbert” space structure on \(\kformsOver{k} \manifold\); for \(\alpha, \beta \in \kformsOver{k}{\manifold}\) we set \[ \scal {\alpha} {\beta} \letDef \int_{\manifold} g(\alpha, \beta) \omega = \int_{\manifold} (\hodge \alpha) \wdg \beta = \int_{\manifold} \alpha \wdg (\hodge \beta). \] Our goal will now be to find the adjoint of the exterior derivative \(\d : \kformsOver{k}{\manifold} \to \kformsOver{k+1}{\manifold}\) with respect to the above-defined scalar product. To this end, given \(\alpha \in \kformsOver{k}{\manifold}\), \(\beta \in \kformsOver{k+1}{\manifold}\) we compute \[\begin{align*} \scal{\d \alpha}{\beta} &= \int_{\manifold} \d \alpha \wdg \beta \\ &= \int_{\manifold} \d (\alpha \wdg (\hodge \beta)) - (-1)^k \int_{\manifold} \alpha \wdg \d(\hodge \beta) \\ &\tOnTop{=}{Stokes} \int_{\boundaryOf \manifold} \alpha \wdg (\hodge \beta) - (-1)^k \int_{\manifold} \alpha \wdg \d(\hodge \beta) \end{align*}\] where we used the fact \(\d (\alpha \wdg (\hodge \beta)) \fromDef \d \alpha \wdg (\hodge \beta) + (-1)^k \alpha \wdg \d (\hodge \beta)\) and a generalization of Stokes’ theorem 3.1 to manifolds. The first term vanishes if \(\manifold\) does not have a boundary, hence by \(\hodge \hodge = (-1)^{n(k-k)} \id\) \[ \begin{aligned} \scal{\d \alpha}{\beta} &= (-1)^{k+1} \int_{\manifold} \alpha \wdg \d (\hodge \beta) \\ &= (-1)^{k+1} (-1)^{k(n-k)} \int_{\manifold} \alpha \wdg \hodge (\hodge \d \hodge \beta) \\ &= (-1)^{k+1 + nk - k^2} \int_{\manifold} (\hodge \alpha) \wdg (\hodge \d \hodge \beta) \\ &\fromDef (-1)^{nk + 1} \scal {\alpha} {\hodge \d \hodge \beta}, \end{aligned} \] where the last equality holds as \(k - k^2 \equiv 0 \mod 2\). Thus, \(\adjd \letDef (-1)^{nk+1} \hodge \d \hodge\) is the adjoint of \(\d\) with respect to the scalar product on \(\kformsOver{k}{\manifold}\).

Henceforth, we have can define the self-adjoint operator \(\lapl \letDef \adjd \d + \d \adjd\) on \(\kformsOver{k}{\manifold}\), which is typically the Laplace operator or Laplacian. In the case of \(k = 0\), it simplifies to \[ \lapl f = \adjd \d f + \d \underbrace{\adjd f}_0 = - \hodge \d \hodge \d f, \] where \(\adjd f = 0\) because \(\adjd : \kformsOver{k}{\manifold} \to \kformsOver{k-1}{\manifold}\) with \(\kformsOver{-1}{\manifold} = \set{0}\). The important takeaway here is that the laplacian exists only if we have a metric. Recall that typically the laplacian is defined as the divergence of gradient \(\d f(\vi v) = \scal {\grad f}{\vi v}\), which, in turn, depends on the scalar product. In other words, this reliance on a scalar product was part of the definition even in \(\R^3\).

Let us now focus on computing the Hodge-star of a 1-form \(\lambda = a_i \d u^i\). In fact, it suffices to compute only \(\hodge \d u^i = b_i^j \d \hat{u}^j\) with \(\d \hat{u}^j = \d u^1 \wedges \cancel{\d u^j} \wedges \d u^n\). Clearly, \[ \hodge \d u^i \wdg \d u^l \fromDef \scal{(-1)^{(k+1)(n-(k+1))} \dd u^i} {\d u^j} \omega \onTop{=}{k=1} \scal {\d u^i} {\d u^j} \omega, \] where the left-hand side expands to sum over \(b_i^j \d \hat{u}^j \wdg \d u^l\) with the only non-zero term for \(j = l\). Moving \(l\) to the correct position produces \[\begin{gather*} (-1)^{n-l} b_i^l \d u^1 \wedges \d u^n = \hodge \d u^i \wdg \d u^l = g^{il} \underbrace{\sqrt{g} \dd u^1 \wedges \d u^n}_{\omega} \\ \Downarrow \\ b_i^l = (-1)^{n-l} \sqrt{g} \implies \hodge \d u^i = \sum_{l} (-1)^{n-l} g^{il} \sqrt{g} \dd \hat{u}^l. \end{gather*}\]

3.4.1 Green’s Formula

Consider now a sub-manifold \(\mcal N \subseteq \manifold\) with \(\dim \mcal N = \dim \manifold\) a two functions (0-forms) \(\vf, \psi\) on \(\manifold\). A natural definition of the \(\mcal N\)-restricted scalar product of functions on \(\mcal N\), i.e., a bilinear form on \(\vf, \psi\), is (by inclusion into \(\manifold\) and) employing the scalar product of \(k\)-forms, \[ \int_{\mcal N} \d \vf \wdg \hodge \dd \psi = \int_{\mcal N} \hodge \dd \vf \wdg \d \psi = \int_{\mcal N} \scal {\d \vf} {\d \psi} \omega. \] Then as \(\d \vf = \pDeriv{\vf}{u^j} \d u^j\) and \(\hodge \d \psi = \hodge \brackets{\pDeriv{\psi}{u^i} \d u^i} = \sum_{l} (-1)^{n-l} g^{il} \sqrt{g} \pDeriv{\psi}{u^i} \d \hat{u}^l\), we get \[\begin{align*} \d \vf \wdg \hodge \dd \psi &= \sum_l (-1)^{n-l} g^{il} \sqrt{g} \pDeriv{\psi}{u^i} \pDeriv{\vf}{u^j} \underbrace{\dd u^j \wdg \d \hat{u}^l}_{0 \iff j \neq} \\ &= (-1)^{n-1} g^{il} \sqrt{g} \pDeriv{\psi}{u^i} \pDeriv{\vf}{u^l} \dd u^1 \wedges \d u^n \\ &= (-1)^{n-1} g^{il} \pDeriv{\psi}{u^i} \pDeriv{\vf}{u^l} \omega, \end{align*}\] which clearly is a quadratic form applied to the derivatives. Before continuing with the main idea of this section, notice that by (3.10) \[ \hodge \overbrace{\d \hodge \dd \psi}^{\text{degree } n} = \lapl \psi \implies \d \hodge \dd \psi = \lapl \psi \omega. \] Furthermore, by the manifold-version of Stokes’ theorem 3.1 we get (keep in mind \(\boundaryOf \mcal N\) is \(n-1\) dimensional space) \[\begin{align*} \int_{\boundaryOf \mcal N} \vf \underbrace{\hodge \dd \psi}_{\text{degree } n-1} &= \int_{\mcal N} \d (\vf \hodge \d \psi) \fromDef \int_{\mcal N} \d \vf \wdg \hodge \dd \psi + \int_{\mcal N} \vf \underbrace{\overbrace{\dd \hodge \d \psi}^{\text{degree } n}}_{\lapl \psi \cdot \omega} \\ &= \int_{\mcal N} \d \vf \wdg \hodge \dd \psi + \int_{\mcal N}\vf \lapl \psi \omega. \end{align*}\] By exchanging the order of \(\vf\) and \(\psi\) in the result above and comparing them, we finally get the Green’s formula, \[ \int_{\boundaryOf \mcal N} \vf \hodge \dd \psi - \psi \hodge \dd \vf = \int_{\mcal N} \brackets{\vf \lapl \psi - \psi \lapl \vf} \omega. \] The main take-home message here is that integration of forms over (oriented) differentiable manifolds generalizes the identities and theorems from classical analysis.

3.4.2 Harmonic Forms and Hodge Decomposition

Definition 3.15 (Harmonic form) A form \(\omega \in \kformsOver{k}{\manifold}\) is called harmonic, if \(\lapl \omega = 0\).

Immediately, we can see that (recall \(\adjd\) is adjoint of \(\d\)) \[ 0 = \scal {\lapl \omega} {\omega} = \scal {(\adjd \d + \d \adjd) \omega} {\omega} = \scal {\d \omega} {\d \omega} + \scal {\adjd \omega} {\adjd \omega} \implies \d \omega = \adjd \omega = 0, \] as the scalar product is, by definition, non-degenerate. Also \(\scal {\d \alpha} {\adjd \beta} = \scal {\d \d \alpha} {\beta} = 0\) using Proposition 3.4, and for \(\lapl \omega = 0\) \[\begin{align*} \scal{\d \alpha} {\omega} = \scal{\alpha} {\adjd \omega} &= 0, \scal {\adjd \beta} {\omega} = \scal {\beta} {\d \omega} &= 0, \end{align*}\] hence the images of \(\d\), \(\adjd\) and the space of harmonic forms are mutually orthogonal. Therefore we can phrase the following statement

Theorem 3.3 (Hodge decomposition) Let \(\eta \in \kformsOver{k}{\manifold}\). Then there exist \(\alpha \in \kformsOver{k-1}{\manifold}\), \(\beta \in \kformsOver{k+1}{\manifold}\), and \(\omega \in \kformsOver{k}{\manifold}\) with \(\lapl \omega = 0\) such that \(\eta = \d \alpha + \adjd \beta + \omega\).

Remark 3.15. Importantly, this composition is unique in the sense that \(\d \alpha\) is unique (however not \(\alpha\), as we could clearly perturb it by any closed form and get the same result). To see that, consider \(0 = \d \alpha + \adjd \beta + \omega\). Applying \(\d\), and \(\adjd\) to it, respectively, gives \[ \rcases{ 0 = \d \d \alpha + \d \adjd \beta + \d \omega & \implies \d \adjd \beta = 0 \\ 0 = \adjd \d \alpha + \adjd \adjd \beta + \adjd \omega & \implies \adjd \d \alpha = 0 } \implies \lcases{ 0 = \scal {\d \adjd \beta} {\beta} = \scal {\adjd \beta} {\adjd \beta} & \implies \adjd \beta = 0, \\ 0 = \scal {\adjd \d \alpha} {\alpha} = \scal {\d \alpha} {\d \alpha} & \implies \d \alpha = 0, } \] and thus also \(\omega = 0\). In other words, this decomposition maps to \(0\) if and only if all the terms are \(0\).

The map \(h : \ker (\lapl) \to H^p(\manifold)\), where \(\ker (\lapl)\) is the space of harmonic forms and the \(p\)-th cohomology space \(H^p(\manifold)\) is the quotient space of closed \(p\)-forms modulo exact \(p\)-forms, with \(\omega \mapsto [\omega]\) is an isomorphism.

As a side-note, also recall that we have an isomorphism between \(H^p(\manifold)\) and \(p\)-chains modulo \(p\)-boundaries.

Proof. Indeed, one can verify that

• \(h\) is injective: if \([\omega] = 0\) then \(\omega\) is an exact form, i.e., \(\omega = \d \alpha\). As \(\omega\) is also harmonic, we get \(\adjd \omega = 0\), hence \[ 0 = \scal {\adjd \omega} {\alpha} = \scal {\omega} {\d \alpha} = \scal {\omega} {\omega} \implies \omega = 0; \]
• \(h\) is surjective: take a representative \(\eta \in [\eta]\), then, by definition, \(\eta\) is closed. Using Theorem 3.3, we get \(\eta = \d \alpha + \adjd \beta + \gamma\), and \[ 0 = \d \eta = 0 + \d \adjd \beta + 0 \implies \d \adjd \beta = 0 \implies \scal {\d \adjd \beta} {\beta} = \scal {\adjd \beta} {\adjd \beta} = 0 \implies \adjd \beta = 0. \] Thus \(\eta = \d \alpha + \gamma\), which, in turn, implies \(\gamma \in [\eta]\) (as they differ up to a closed form \(\omega = \d \alpha\)).

In particular, this shows that \(p\)-th cohomology space \(H^p(\manifold)\) is finite dimensional (although both “original” spaces are infinite dimensional), as it is isomorphic to a finite dimensional space.

3.5 A (Little) Bit on Riemannian Geometry

Consider again a differentiable manifold \(\manifold\) with the tangent spaces \(\tangentAt{P} \manifold\) forming the tangent bundle \(\tangent \manifold\). Moreover, let \(\vi X\) a vector field, and recall that vector fields act on functions by \(\vi X f\) (as directional derivatives), see (3.13).

Let now \(\vi X, \vi Y\) be vector fields, then as we have discussed in Remark 3.9, \(\vi Y \vi X\) is not a vector field, i.e., it does not coincide with directional derivatives. To remedy this, we have introduced the Lie bracket \([\vi X, \vi Y] = \vi X \vi Y - \vi Y \vi X\), which was again a vector field. Nonetheless, we would like to also have directional derivatives of vector fields, as the naive construction above clearly does not work/suffice.

Definition 3.16 (Connection) A map \(\connect : \tangent \manifold \times \text{vector fields} \to \tangent \manifold\) is called a connection, if it satisfies

1. (linearity in direction): \(\connect_{\alpha \xi + \beta \eta} \vi X = \alpha \connect_{\xi} \vi X + \beta \connect_{\eta} \vi X\) for every point \(P \in \manifold\) and \(\xi, \eta \in \tangentAt{P} \manifold\) and
2. (function-vector field product rule): \(\connect_{\xi} f \vi X = \xi f \cdot \vi X + f \connect_{\vi \xi} \vi X\) at every point \(P\) such that¹³ \(\xi \in \tangentAt{P} \manifold\) .

Remark 3.16. In coordinates we have \(\connect_{\partialOp{u^i}} \partialOp{u^j} = \christoffel^l_{ij} \partialOp{u^l}\), where \(\christoffel^l_{ij}\) are called the Christoffel symbols (or the connection coefficients).

Example 3.8 Take for simplicity \(\R^3\) and manifold \(\manifold\) prescribed by \(\vi F : U \to \R^3\) such that \(U \subseteq \R^2\) and assume \(\pDeriv{\vi F}{u^1}\), \(\pDeriv{\vi F}{u^2}\) are linearly independent for all \(\vi u = (u^1, u^2) \in U\) (see Figure 3.5 for illustration). Then \[ \underbrace{\frac {\partial^2 \vi F} {\partial u^1 \partial u^2}}_{\in \R^3} = \underbrace{\christoffel^l_{ij} \pDeriv {\vi F} {u^l}}_{\in \tangentAt{P} \manifold} + h_{ij} \vi n, \] where \(\vi n\) is a normal vector such that \(\norm{\vi n} = 1\). Clearly, \(h_{ij}\) holds the curvature information.

Figure 3.5: Differential manifold and a parametrized curve in \(\R^3\)

Now consider a parametrized curve \(\curve(t)\) in \(\manifold\), which reads \(\curve(t) = \vi F \after \gamma(t)\). For \(\gamma_i(t) = u^i(t)\) we have \(\dot{\curve}(t) = \pDeriv{\vi F}{u^i} \dot{u}^i(t)\). To get the curvature, we need second derivatives, i.e., \[ \ddot{\curve}(t) = \frac{\partial^2 \vi F}{\partial u^i \partial u^j} \dot{u}^i(t) \dot{u}^j(t) + \pDeriv{\vi F}{u^i} \ddot{u}^i(t), \] where we assumed that \(\curve(t)\) was parametrized by arc-length (more on this later). The curvature is then \(\scal {\ddot{\curve}} {\vi n} = h_{ij} \dot{u}^i \dot{u}^j\).

In the general case, consider a curve \(\curve\) on \(\manifold\) given by parametrization \(\gamma\) in a chart, i.e., \(\gamma : u^i = u^i(t)\) for \(t \in [a,b]\). Then \(\vi X \after \curve(t)\) is the vector field along the curve \(\curve\). Moreover, let \(T(t)\) be the tangent vector of \(\curve\), i.e., \(T(t) = \pDeriv{u^i}{t} \partialOp{u^i} \in \tangentAt{\curve(t)} \manifold\). Then for \(\vi X = x^i \partialOp{u^i}\) we have (by Remark 3.16) \[ \connect_T \vi X \tOnTop{=}{linear} T x^i \partialOp{u^i} + x^i \connect_T \partialOp{u^i} = \pDeriv{x^i}{u^j} \pDeriv{u^j}{t} \partialOp{u^i} + x^i \underbrace{\christoffel^k_{ij} \pDeriv{u^j}{t} \partialOp{u^i}}_{\text{in coordinates}}. \tag{3.25}\]

Definition 3.17 (Parallel vector field) A vector field \(\vi X\) is called parallel along the curve \(\curve\), if \(\connect_T \vi X = 0\) for all \(t \in [a,b]\).

Note that the definition of a parallel vector field to a given curve does not depend on parametrization of the curve. Moreover, from (3.25) follows that for a given curve \(\curve\) (with parametrization by \(u^i(t)\)) a parallel vector field solves (as \(\pDeriv{x^i}{u^j} \pDeriv{u^j}{t} = \deriv{x^i}{t}\)) \[ \connect_T \vi X = \brackets{\deriv{x^k}{t} + x^i \christoffel^k_{ij} \pDeriv{u^j}{t}} \partialOp{u^k} = 0, \tag{3.26}\] which is a linear system of ODEs for the coordinate functions \(x^k = x^k(t)\) of the vector field \(\vi X\) (along the curve \(\curve\)). Classical results from the theory of linear ODE systems imply that for a fixed choice of \(\vi X(a) \in \tangentAt{\gamma(a)} \manifold\) we have a unique solution, and that linearly independent choice of \(\vi X(a)\) yield linearly independent solutions for all \(t\). Hence, \(\vi X(a) \in \tangentAt{\gamma(a)} \manifold \mapsto \vi X(b) \in \tangentAt{\gamma(b)} \manifold\) is an isomorphism of tangent spaces!

In other words, as soon as have defined \(\connect_{\xi}\) (which depends on the curve), we can connect tangent spaces by these isomorphism. In fact, this is where the name “connection” comes from.

Remark 3.17. Let us repeat that the connection isomorphism depends on the curve \(\curve\), however not on its parametrization.

A special, but very important, case of a parallel vector field to a given curve arises when the curve is parallel to itself! These curves are called geodesics, and serve as generalizations of straight lines to a “curved” manifold \(\manifold\). We say that geodesic curves parallely transport their own tangents.

Definition 3.18 (Geodesic curve) A curve \(\curve\) is called geodesic if its tangent vector \(T(t)\) satisfies \(\connect_{T} T = 0\).

A curve \(\curve\) induces a vector field (defined only along \(\curve\)) via its tangent \(T(t)\), i.e., \(x^k = \pDeriv{u^k}{t} = \dot{u}^k\). Then the linear ODE system then give(3.26) becomes \[ \ddot{u}^k + \christoffel^k_{ij} \dot{u}^i \dot{u}^j = 0, \tag{3.27}\] which is, however, non-linear 2nd order system of ODEs.

Let, for simplicity, \(a = 0\) and \(b = 1\). Choosing initial conditions \(P = \gamma(0)\) (i.e., also \(u^i(0)\) give \(P\)) and \(T(0) = \vi \xi \in \tangentAt{P} \manifold\) suffices to provide a unique solution \(of u^i(t)\), thus to unique prescribe \(\gamma_{\vi \xi}(t)\).

In total, this gives an exponential map \(\exp_P : \tangentAt{P} \manifold \to \manifold\) with \(\vi \xi \mapsto \gamma_{\vi \xi}(1)\). Intuitively, the exponential map \(\exp_P\) gives us points connected to \(P\) by straight lines in \(\manifold M\). Note that as \(\tangentAt{P} \manifold\) is a vector space, straight lines take the usual notion.

3.5.1 Riemannian Manifolds

Assume from now on that we have a Riemannian metric on \(\manifold\), i.e., a positive definite bilinear form \(g : \tangentAt{P} \manifold \times \tangentAt{P} \manifold \to \R\) on all tangent spaces, and set \(g(\vi X, \vi Y) =: \scal {\vi X} {\vi Y}\) as the scalar product.

Definition 3.19 (Levi-Civita connection) A connection \(\connect\) is called Levi-Civita connection, if it satisfies for all vector fields \(\vi X, \vi Y, \vi Z\) the following properties:

1. \(\connect_{\vi X} \vi Y - \connect_{\vi Y} \vi X = \lieBracket{\vi X}{\vi Y}\);
2. \(\vi Z \scal {\vi X} {\vi Y} = \scal{\connect_{\vi Z} \vi X}{\vi Y} + \scal{\vi X}{\connect_{\vi Z} \vi Y}\).

Proposition 3.6 A Levi-Civita connection \(\connect\) is given uniquely by the required properties.

Proof. Firstly, notice that giving \(\scal{\connect_{\vi X} \vi Y} {\vi Z}\) for any \(\vi X, \vi Y, \vi Z\) uniquely determines \(\connect_{\vi X} \vi Y\). Thus \[\begin{align*} \scal{\connect_{\vi X} \vi Y} {\vi Z} &\tOnTop{=}{2.} \vi X \scal {\vi Y} {\vi Z} - \underbrace{\scal {\vi Y} {\connect_{\vi X} \vi Z}} \tOnTop{=}{1.} \vi X \scal{\vi Y} {\vi Z} - \underbrace{\scal{\vi Y} {\connect_{\vi Z} \vi X}} - \scal{\vi Y} {\lieBracket{\vi X}{\vi Z}} \\ &\tOnTop{=}{2.} \vi X \scal{\vi Y} {\vi Z} - \vi Z \scal{\vi X} {\vi Y} + \underbrace{\scal{\vi X} {\connect_{\vi Z} \vi Y}} - \scal{\vi Y} {\lieBracket{\vi X}{\vi Z}} \\ &\tOnTop{=}{1.} \vi X \scal{\vi Y} {\vi Z} - \vi Z \scal{\vi X} {\vi Y} + \underbrace{\scal{\vi X} {\connect_{\vi Y} \vi Z}} - \scal {\vi X} {\lieBracket{\vi Y}{\vi Z}} - \scal{\vi Y} {\lieBracket{\vi X}{\vi Z}} \\ &\tOnTop{=}{2.} \vi X \scal{\vi Y} {\vi Z} - \vi Z \scal{\vi X} {\vi Y} + \vi Y \scal{\vi X} {\vi Z} - \underbrace{\scal{\connect_{\vi Y} \vi X}{\vi Z}} - \scal {\vi X} {\lieBracket{\vi Y}{\vi Z}} - \scal{\vi Y} {\lieBracket{\vi X}{\vi Z}} \\ &\tOnTop{=}{1.} \vi X \scal{\vi Y} {\vi Z} - \vi Z \scal{\vi X} {\vi Y} + \vi Y \scal{\vi X} {\vi Z} - \scal{\connect_{\vi X} \vi Y} {\vi Z} + \scal {\vi Z} {\lieBracket{\vi X}{\vi Y}} - \scal {\vi X} {\lieBracket{\vi Y}{\vi Z}} - \scal{\vi Y} {\lieBracket{\vi X}{\vi Z}}, \end{align*}\] hence \[ \scal {\connect_{\vi X} \vi Y} {\vi Z} = \frac 1 2 \brackets{ \vi X \scal {\vi Y} {\vi Z} + \vi Y \scal{\vi X} {\vi Z} - \vi Z \scal{\vi X} {\vi Y} - \scal {\vi X} {\lieBracket{\vi Y} {\vi Z}} - \scal {\vi Y} {\lieBracket{\vi X} {\vi Z}} + \scal {\vi Z} {\lieBracket{\vi X}{\vi Y}} }. \tag{3.28}\] As the right-hand side of said identity does not depend on \(\connect\), we have successfully shown that a Levi-Civita connection is given uniquely. In other words, if we want the connection \(\connect\) to be Levi-Civita, we must necessarily use (3.28).

Let us now use (3.28) to compute the Christoffel symbols \(\christoffel^k_{ij}\). Consider the following (chosen) setting \[ \vi X = \partialOp{u^i}, \; \vi Y = \partialOp{u^j}, \; \vi Z = \partialOp{u^l} \implies \connect_{\vi X} \vi Y \fromDef \christoffel^k_{ij} \partialOp{u^k}. \] Then by Remark 3.9 we have \(\lieBracket {\vi X}{\vi Y} = \lieBracket {\vi Y} {\vi Z} = \lieBracket{\vi Z} {\vi X} = 0\) and \[ \scal{\connect_{\vi X} \vi Y} {\vi Z} = \christoffel^k_{ij} g_{kl} \tOnTop{=}{by above} \frac 1 2 \brackets{\partialOp{u^i} g_{jl} + \partialOp{u^j} g_{il} - \partialOp{u^l} g_{ij}}, \] thus \[ \christoffel^k_{ij} = \frac 1 2 g^{kl} \brackets{\partialOp{u^i} g_{jl} + \partialOp{u^j} g_{il} - \partialOp{u^l} g_{ij}}. \tag{3.29}\] Hence also \(\christoffel^k_{ij} = \christoffel^k_{ji}\) as a consequence of property 1. of Definition 3.19. Unless noted otherwise, we will assume we are working with the Levi-Civita connection (induced by the chosen scalar product).

Previously, we have shown that from connections we get the (notion of) parallel transport 3.17 of vector fields along curves. Moreover, we have discussed how vector fields parallel to curves provide isomorphism of tangent spaces at points along this curve. As we are now in the Riemannian manifold setting (i.e., we have a scalar product), we strengthen this result to isometries.

Proposition 3.7 Parallel transport along curves induces isometries on tangent spaces.

Proof. Let \(\curve\) be a curve given by a parametrization \(\gamma\) and \(T = \dot{\gamma}\) is tangent vector. Let \(\vi X\) be a vector field parallel to \(\curve\), i.e., \(\connect_{T} \vi X = 0\). By the 2nd property of Definition 3.19 if \(\vi X\) and \(\vi Y\) are parallel along \(\curve\), then \[ T \scal{\vi X} {\vi Y} = \scal{\connect_{T} \vi X}{\vi Y} = \scal {\vi X} {\connect_T \vi Y} = 0 \] implies \(\scal {\vi X} {\vi Y}\) is constant along \(\curve\) (as it has zero tangent-directional derivative at all points of the curve). Hence the isomorphism also preserves the Riemannian metric, i.e., it is actually an isometry.

Remark 3.18. Recall Definition 3.18, which says that a curve \(\curve\) given by \(\gamma\) is called a geodesic, if \(\connect_T T = 0\), where \(T\) is its tangent vector. This, by Proposition 3.7, implies that \(\norm{T}\) is constant along \(\curve\). Without a significant restriction we can require \(\norm{T} = 1\).

So far, we have only mentioned geodesics in the sense that they parallely transport its own tangent. However, it turns out they also provide, in a certain sense, shortest paths.

Definition 3.20 (Arclength) Let \(\curve\) be a curve given by a parametrization \(\gamma : u^i = u^i(t)\) for \(t \in [a,b]\). We define the length of the tangent as¹⁴ \[ \d s^2 = g_{ij} \d u^i \d u^j = g_{ij} \dot{u}^i \dot{u}^j \dd t^2, \] which, in turn, prescribes the arclength of \(\curve\) as \[ \lengthOf{\curve} = \int_a^b \dd s = \int_a^b \sqrt{g_{ij} \dot{u}^i \dot{u}^j} \dd t. \]

Our goal will now be to find curves \(\curve\) that minimize \(\lengthOf{\curve}\) for given end-point of the curve, see Figure 3.6.

Figure 3.6: Illustration of a geodesic curve \(\curve\) and a non-geodesic \(\curve'\) between points \(P\) and \(Q\)

Let us define a displaced curve \(\curve_{\ve}\) with parametrization \(\gamma_{\ve}(t)\) by \(u^i = u^i(t) + \ve h^i(t)\) with \(h^i(a) = h^i(b) = 0\) for \(i \in \oneToN{n}\). We will seek curves such that \[ \evaluateAt{\derivOp{\ve} \lengthOf{\curve_{\ve}}}{\ve = 0} = 0, \] i.e., they are stationary points with respect to displacement. Now by Definition 3.20 \[ \lengthOf{\curve_{\ve}} = \int_a^b \sqrt{g_{ij} (\dot{u}^i + \ve \dot{h}^i) (\dot{u}^j + \ve \dot{h}^j)} \dd t, \] where importantly \(g_{ij}\) also depend on \(\ve\) and \(h^i\) (as we move \(u^i\) by the displacements, and thus change the tangent space, which carries \(g_{ij}\)), and we shall assume \(g_{ij} \dot{u}^i \dot{u}^j = 1\) (which is just a change of variables). By viewing the derivative as a linearization, we get (by our assumption) \[\begin{align*} \sqrt{g_{ij} (\dot{u}^i + \ve \dot{h}^i) (\dot{u}^j + \ve \dot{h}^j)} &= \sqrt{1 + \ve \brackets{ 2 g_{ij} \dot{u}^i \dot{h}^j + \partialOp{u^l} g_{ij} h^l \dot{u}^i \dot{u}^j } + \HOT} \\ &= 1 + \frac {\ve} 2 \brackets{ 2 g_{ij} \dot{u}^i \dot{h}^j + \partialOp{u^l} g_{ij} h^l \dot{u}^i \dot{u}^j } + \HOT \end{align*}\] where the second equality comes from linearization of \(\sqrt{x}\) to \(\frac x 2\). Then \[ \evaluateAt{\derivOp{\ve} \lengthOf{\curve_{\ve}}}{\ve = 0} = \frac 1 2 \int_a^b \brackets{ 2 g_{ij} \dot{u}^i \dot{h}^j + \partialOp{u^l} g_{ij} h^l \dot{u}^i \dot{u}^j } \dd t \EQ 0 \] for all choices of \(h^i\) such that the boundary conditions are satisfied. Notice that the first term of the integrand depends on \(\dot{h}^j(t)\), whereas the second term only on \(h^l(t)\), which suggest integration by parts. Indeed \[\begin{align*} \derivOp{\ve} \lengthOf{\curve_{\ve}} &= \frac 1 2 \bigg( \underbrace{\evaluateAt{2 g_{il} \dot{u}^i h^l}{t = a}^b}_{h^l(a) = h^l(b) = 0} - \int_a^b \derivOp{t} (2 g_{il} \dot{u}^i) h^l \dd t + \int_a^b \partialOp{u^l} g_{ij} h^l \dot{u}^i \dot{u}^j \dd t \bigg) \\ &= \frac 1 2 \int_a^b \brackets{- 2 \pDeriv{u^j} g_{il} \dot{u}^i \dot{u}^j - 2 g_{il} \ddot{u}^i + \partialOp{u^l} g_{ij} \dot{u}^i \dot{u}^j} h^l \dd t \EQ 0. \end{align*}\] Now for this to vanish for all choices of \(h^l\), the term in the bracket must vanish identically. Interestingly, this is where the theory of distributions 2 comes from, as the \(h^l\) plays a role of the test functions and want the terms in brackets to be an identically zero distributions. In total, we now have \[ g_{il} \ddot{u}^i + \frac 1 2 \partialOp{u^j} g_{il} \dot{u}^i \dot{u}^j + \frac 1 2 \partialOp{u^i} g_{jl} \dot{u}^i \dot{u}^i - \frac 1 2 \partialOp{u^l} g_{ij} \dot{u}^i \dot{u^l} = 0, \] which we can multiply by the inverse matrix \(g^{lk}\) to isolate the first term, yielding \[ \ddot{u}^k + \underbrace{\frac 1 2 g^{lk} \brackets{ \partialOp{u^j} g_{il} + \partialOp{u^i} g_{jl} - \partialOp{u^l} g_{ij} }}_{\fromDef \christoffel_{ij}^k} \dot{u}^i \dot{u}^j = 0. \] This finally results in the differential equation \[ \ddot{u}^k + \christoffel^k_{ij} \dot{u}^i \dot{u}^j = 0, \tag{3.30}\] which by (3.27) is equivalent with \(\connect_T T = 0\). Under our assumptions on \(g_{ij}\) (and, by extension, \(\christoffel^k_{ij}\) via the Levi-Civita connection) for any initial data¹⁵ \(u^i(0) = P^i\) and \(\dot{u}^i = \xi^i\) there exists and interval \((\underbrace{t_{\min}}_{<0}, \underbrace{t_{\max}}_{>0})\) such that the solution \(u^i(t)\) of (3.30) exists on this interval. For \(\vi \xi \in \tangentAt{P} \manifold\) define the function \(\gamma_{\vi \xi}(t)\) as the unique solution satisfying \(\gamma_{\vi \xi}(0) = P\) and \(\dot{\gamma}_{\vi \xi}(0) = \vi \xi\), then \(\gamma_{t \vi \xi}(s) = \gamma_{\vi \xi} (ts)\) (when everything exists).

Moreover, these results can be used to define an exponential map \(\exp_P : U \to \manifold\), where \(U\) is an open neighborhood of \(\vi 0 \in \tangentAt{P} \manifold\), by \(\vi \xi \mapsto \gamma_{\vi \xi}(1)\), see Figure 3.7 a).

Figure 3.7: Illustration of the exponential map \(\exp_P\) in both local and global perspectives

Importantly, while geodesics themselves define straight lines in a global manner, interpreting them as shortest path is necessarily possible only via a local viewpoint (as should be hinted by the fact that we only required them to be stationary points with respect to displacements, not global minimizers). If, for example, we take Earth and move along a meridian from the north pole to the south pole and even a bit further (i.e., we “overshoot” the south pole), then such geodesic is surely longer than if we went from the north pole the other way around. For illustration see Figure 3.7 b), where both curves plotted are geodesics. Furthermore, we can use geodesics to define a metric space structure on \(\manifold\).

Proposition 3.8 Take a curve \(\omega : [0, b] \to \manifold\) such that there exist \(0 = t_0 < t_1 < \dots < t_n = b\) with \(\evaluateAt{\omega}{(t_i, t_{i+1})} \in \Contf{\infty}\) satisfying \(\connect_{\dot{\omega}} \dot{\omega} = 0\). We call such an \(\omega\) a piecewise geodesic. Using this notion, we define a metric on \(\manifold\) as \[ d(P,Q) \letDef \inf \set{\lengthOf{\omega} \divider \omega(0) = P \land \omega(b) = Q}. \tag{3.31}\]

Proof. The function in (3.31) is surely a metric, as the following properties hold:

• \(d(P, Q) = d(Q, P)\), which follows from parametrizing the minimizing piecewise geodesic in the opposite dimension;
• \(d(P, Q) \geq 0\);
• \(d(P, R) \leq d(P, Q) + d(Q, R)\), where as we fix \(Q\) we effectively take 2 infima over smaller sets (thus both of them can only be larger).

Lastly, we need to show that \(d(P, Q) = 0 \implies P = Q\). To this end, recall that \(g\) (the bilinear form underlying our scalar product on \(\tangentAt{P} \manifold\)) is positive definite, thus for a compact subset \(K\) of \(U\) we have that \(\norm{\vi \xi}_{\tangentAt{P} \manifold} \geq \lmbd \norm{\d \vf_P(\vi \xi)}_{\R^n}\) for the corresponding chart map \(\vf\). Let now \(\omega : [0, b] \to \manifold\) be any piecewise \(\Contf{1}\) curve (namely any piecewise geodesic) staying in \(U\). Its arclength is then by Definition 3.20 \[ \lengthOf{\omega} = \int_0^b \sqrt{g_{ij} \dot{u}^i \dot{u}^j} \dd t = \int_0^b \norm{\dot{\omega}(t)}_{\tangentAt{\omega(t)}\manifold} \dd t \geq \lmbd \int_0^b \norm{\d \vf_{\omega(t)}(\dot{\omega}(t))}_{\R^n} \dd t = \lmbd \int_0^b \norm{\derivOp{t} (\vf \after \omega)(t)}_{\R^n} \dd t, \] and hence we’re in the usual Euclidean setting. Here, we directly get \[ \lengthOf{\omega} \geq \lmbd \int_0^b \norm{\derivOp{t} (\vf \after \omega)(t)}_{\R^n} \dd t \geq \lmbd \norm{\vf(\omega(b)) - \vf(\omega(0))}_{\R^n} = \lmbd \norm{\vf(Q) - \vf(P)}_{\R^n}, \] hence the desired implication.

Definition 3.21 (Geodesically complete manifold) We call a manifold \(\manifold\) geodesically complete, if \(\exp_P\) exists for all \(\vi \xi \in \tangentAt{P} \manifold\) and all \(P \in \manifold\).

Remark 3.19. In a geodesically complete manifold, we can view the exponential map \(\exp\) as a mapping \(\manifold \times \tangent \manifold \to \manifold\) such that \((P, \vi \xi) \mapsto \exp_P(\vi \xi)\).

Before we introduce (and prove) the two Rinov-Hopf theorems concluding this course, we shall state a supplementary result, which provides a local description of the metric using geodesics.

Proposition 3.9 Let \(\manifold\) be a Riemannian manifold. Then for every \(P \in \manifold\) there exists \(\ve(P) >0\) and a neighborhood \(U\) of \(P\) such that:

1. any two points in \(U\) are connected by a geodesic of length \(< \ve(P)\), and
2. for all \(Q \in U\) we have that \(\evaluateAt{\exp_Q}{\ball{\vi 0}{\ve(P)}}\) is a diffeomorphism.

Theorem 3.4 (Rinov-Hopf on minimal geodesics) Let \(\manifold\) be a connected and geodesically complete manifold. Then any pair of points on \(\manifold\) can be connected by a minimal geodesic¹⁶.

Let us note that this minimal geodesic is not necessarily unique (for example, there is no unique minimal geodesic connection the north and the south pole of Earth).

Proof. Let \(P, Q \in \manifold\), \(P \neq Q\) and \(\delta = d(P, Q) = \inf_{\omega \in \mathrm{Paths}(P,Q)} \lengthOf{\omega}\), where \(\mathrm{Paths}(P,Q)\) denotes the set all paths connecting \(P\) and \(Q\). By Proposition 3.9 take \(\ve(P)\) and \(0 < \delta_0 < \min(\ve, \delta)\). Further consider \(S(P, r) = \set{S \in \manifold \divider d(S, P) = r}\) a sphere of radius \(r\) (in the geometry of \(\manifold\)). For illustration see Figure 3.8.

Take now \(P_0 \in S(P, \delta_0)\) such that \(d(P_0, Q) = d(S(P, \delta_0), Q)\), i.e., the element of the sphere \(S(P, \delta_0)\) that is closest to \(Q\). By the definition of \(\ve\) (Proposition 3.9 2.) we can take \(\vi \xi = \frac 1 {\delta_0} \exp_P\Inv(P_0) \in \tangentAt{P} \manifold\), which we can do as \(\exp_{P}\) is diffeomorphic \(\ve\)-close to \(P\), then \(\norm{\vi \xi} = 1\). We will show that \(d(\gamma_{\vi \xi}(t), Q) = \delta - t\) for all \(0 \leq t \leq \delta\), which would prove the theorem. Note that we already defined \(\gamma_{\vi \xi}(t)\) to be distance \(t\) from \(P\).

We shall first prove the statement for \(t = \delta_0\). Then by Proposition 3.8 \[ \delta = d(P,Q) \leq \underbrace{d(P, P_0)}_{\delta_0} + d(P_0, Q) = \delta_0 + d(P_0, Q). \] Let \(\omega : [0,1] \to \manifold\) be any path connecting \(P\) and \(Q\) with \(\omega(0) = P\) and \(\omega(1) = Q\). Then there exists \(\alpha\) such that \(\omega(\alpha) \in S(P, \delta_0)\) and \[ \lengthOf{\omega} = \lengthOf{\omega[0, \alpha]} + \lengthOf{\omega[\alpha, 1]} \geq \delta_0 + d(\omega(\alpha), Q) \geq \delta_0 + d(P_0, Q) \geq \delta, \] as \(P_0\) was defined as the closest element of \(S(P_0, \delta_0)\) to \(Q\). Now, recalling \(P_0 = \gamma_{\vi \xi}(\delta_0)\), we get \[ \delta \fromDef \inf_{\omega} \lengthOf{\omega} \geq \delta_0 + d(P_0, Q) \geq \delta \implies d(\gamma_{\vi \xi}(\delta_0), Q) = \delta - \delta_0. \] Moreover, we can repeat this argument for \(t < \delta_0\) by cutting the path again, i.e., it yields that \(d(\gamma_{\vi \xi}(t), Q) = \delta - t\) for \(t \in [0, \delta_0]\) (such that it holds on the closed interval by continuity).

Assume now there is a maximal \(\delta_1 = \maxOf{t \in [0, \delta] \divider d(\gamma_{\vi \xi}(t), Q) = \delta - t} < \delta\) such that the relation holds. We shall show this leads to a contradiction. If \(\delta_1 < \delta\) then there exist \(P_1 = \gamma_{\vi \xi}(\delta_1)\) and take \(0 < \delta_2 < \min(\delta - \delta_1, \ve(P_1))\). Analogously to \(P_0\), take \(P_2 \in S(P_1, \delta_2)\) such that \(d(P_2, Q) = d(S(P_1, \delta_2), Q)\). We can now repeat the same idea as above for \(P_1, P_2\) and \(Q\) to produce \[ \delta_2 + d(P_2, Q) = \delta - \delta_1 \implies d(P_2, Q) = \delta - (\delta_1 + \delta_2), \tag{3.32}\] thus we only need to make sure nothing broke at \(P_1\) (i.e., that we are not using a piecewise geodesic). As \(\delta_2 < \ve(P_1)\), where \(\exp_{P_1}\) is diffeomorphic, \(P_1\) and \(P_2\) are connected by a unique geodesic, which we can view, by geodesic completeness of \(\manifold\), as the extension of \(\gamma_{\vi \xi}\) (this extension together with \(\gamma_{\vi \xi}[0, \delta_1]\) is still a geodesic). This in combination with (3.32) contradicts the maximality of \(\delta_1\).

Figure 3.8: Illustration of a minimal geodesic \(\gamma_{\vi \xi}\) and the construction of the proof of Theorem 3.4

Theorem 3.5 (Rinov-Hopf on geodesical completeness) If \(\manifold\) is a complete metric space, then \(\manifold\) is geodesically complete.

This theorem can be re-phrased such that the only way for geodesics to end is to “fall off flat earth”, i.e., to run outside their domain.

Proof. Let \(\manifold\) be a complete metric space. Assume there exists \(P \in \manifold\), \(\vi \xi \in \tangentAt{P} \manifold\) such that \(\gamma_{\vi \xi}(t)\) exists for all \(\alpha < t < \beta\), where \(\alpha < 0\) and \(\beta > 0\). Without loss of generality, assume \(\alpha > -\infty\). Take \(t_n\) a decreasing sequence to \(\alpha\), then for \(m > n\) \[ d(\gamma_{\vi \xi}(t_n), \gamma_{\vi \xi}(t_m)) \leq \lengthOf{\gamma_{\vi \xi}([t_n, t_M])} = \norm{\vi \xi} \cdot \absval{t_m - t_n} \] by Definition 3.20 (and that \(u^i = \xi^i\)). Hence \(\gamma_{\vi \xi}(t_n)\) is a Cauchy sequence, which, by completeness of \(\manifold\), attains a limit \(Q = \lim_{n \to \infty} \gamma_{\vi \xi}(t_n) \in \manifold\). As \(\dot{\gamma}_{\vi \xi}(t_n) \in \tangentAt{\gamma_{\vi \xi}(t_n)}\) then by Proposition 3.7 \(\norm{\dot{\gamma}_{\vi \xi}(t_n)} = \norm{\vi \xi}\) (though one should be careful here, as both norms are with respect to different tangent spaces). Along with continuity of the norm and this isometry, it implies that in the tangent bundle \(\tangent \manifold\) we are in a compact subset. Hence there is a subsequence \(\tau_n\) such that \(\dot{\gamma}_{\vi \xi}(t_n) \to \vi \eta\). Now consider the geodesic \(\gamma_{\vi \eta}\) emanating from \(Q\). Then these geodesics agree, i.e., \(\gamma_{\vi \eta}(t - \alpha) = \gamma_{\vi \xi}(t)\), and as \(\gamma_{\vi \eta}\) can be continued, it contradicts the choice of \(\alpha\).

[1]

Lee, J. M. (2012). Introduction to smooth manifolds. Springer New York.

[2]

Munkres, J. R. (1994). Analysis on manifolds. Addison-Wesley, Reading, Mass. [u.a.].

---

1. As a reminder, \(S_k\) denotes the symmetric group of permutations of \(\oneToN{k}\).

2. This is also the reason why we may write \(\norm{\vi e_1 \wdg \dots \wdg \vi e_n} = 1\).

3. In fact, we have already used it when taking about orientation!

4. A function \(f : X \to Y\) between two topological spaces is a homeomorphism if \(f\) is a continuous bijection and the inverse is also continuous, i.e., \(f\) is continuously invertible.

5. A function \(f : X \to Y\) between two topological spaces is a diffeomorphism if \(f\) and its inverse \(f\Inv\) are both differentiable.

6. In case of the differential forms, the coordinate functions \(\omega_i\) are by assumption smooth everywhere on \(\manifold\), hence this is always satisfied.

7. While we do need coordinates to write down \(\dd f\) from (3.13), it is defined coordinate-free by a \(\tangent \manifold\)-section, see Definition 3.6.

8. These determinants can, in fact, be used to construct a form prescribed by them.

9. Hence, for example, the Möbius strip is not orientable.

10. Beware that I found that simplex \(S\) is sometimes defined with \(\sum_{i = 1}^n t^i \leq 1\), whereas other times with \(\sum_{i = 1}^n t^i = 1\).

11. A \(p\)-chain is a chain of \(p\) simplices.

12. Let us note \(\kformsOver{n}{\manifold}\) for \(n\)-dimensional manifold \(\manifold\) is a one-dimensional space, hence we have effectively two choices for the volume form.

13. Recall that \(\xi f\) is the directional derivative of \(f\) in the direction of \(\xi\) at \(P\).

14. Keep in mind that \(\d s^2\) is not a form!

15. Note that we can move from \([a,b]\) to a time interval starting at \(0\) simply by scaling the time.

16. In other words, we do not need the piecewise geodesics to attain the minimum in the metric distance between two points.