\documentclass[11pt]{article} \usepackage{url} %version 1.0,\ \ 09.04.2018 %version 2.0,\ \ 14.04.2018, lots of serious errors % about Jacobi fields (Ivan Frolov) \newcommand{\version}{version 2.0,\ \ 14.04.2018} \newcommand{\firstdate}{11.04.2018} \addtolength{\topmargin}{-5mm} \addtolength{\textheight}{10mm} \addtolength{\oddsidemargin}{-5mm} \addtolength{\textwidth}{10mm} \input{defs-listki-en.tex} \setlength{\headheight}{15pt} \pagestyle{fancy} \lhead{\tiny Hogde theory, HSE} \lfoot{\tiny Issued \firstdate} \cfoot{-- \thepage \ -- } \rfoot{\tiny \sc\version} \rhead{{\tiny Misha Verbitsky}} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \listok{11}{Hodge theory 11: exponential map and Frobenius theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\scriptsize {\bf Rules:} You may choose to solve only ``hard'' exercises (marked with !, * and **) or ``ordinary'' ones (marked with ! or unmarked), or both, if you want to have extra stuff to work. To have a perfect score, a student must obtain (in average) a score of 10 points per week. If you have got credit for 2/3 of ordinary problems or 2/3 of ``hard'' problems, you receive $6t$ points, where $t$ is a number depending on the date when it is done. Passing all ``hard'' or all ``ordinary'' problems brings you $10t$ points. Solving of ``**'' (extra hard) problems is not obligatory, but each such problem gives you a credit for 2 ``*'' or ``!'' problems in the ``hard'' set. The first 3 weeks after giving a handout, $t=1.5$, between 21 and 35 days, $t=1$, and afterwards, $t=0.7$. The scores are not cumulative, only the best score for each handout counts. } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Exponential map} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $\nabla:\; TM \arrow TM \otimes \Lambda^1 M$ be a connection. {\bf Geodesic} with respect to $\nabla$ is a map $\gamma:\; [a, b]\arrow M$ which satisfies $\nabla_{\dot \gamma}(\dot \gamma)=0$, where $\dot \gamma(t)=\frac{\gamma(u)}{du}\restrict {u=t} \in TM\restrict {\im \gamma}$ is the derivative of $\gamma$, and the equation $\nabla_{\dot \gamma}(\dot \gamma)$ is understood as parallelism of $\dot \gamma$, considered as a section of the vector bundle $TM\restrict {\im \gamma}$ with connection $\nabla$. \ed \exercise Let $v\in T_x M$. \enum \ite[!] Prove that for $\epsilon$ sufficiently small, there exists a geodesic $\gamma:\; [0, \epsilon]\arrow M$ such that $\gamma(0)=x$ and $\dot\gamma(0)=v$. \ite[!] Prove that geodesic is unique. \ee \ez \definition Let $U\subset T_x M$ be the set of tangent vectors such that for any $v\in U$ there exists a geodesic $\gamma_v:\; [0, 1]\arrow M$, such that $\gamma_v(0)=x$ and $\dot\gamma_v(0)=v$. The map $v \arrow \gamma(1)$ is called {\bf exponential map}. \ed \exercise[!] Prove that exponential map defines a diffeomorphism from a neighbourhood of 0 in $U$ to $M$. \ez \definition The corresponding coordinates are called {\bf geodesic coordinates}, or {\bf normal coordinates}. \ed \exercise Suppose that $M$ is a Riemannian manifold, and $\nabla$ an orthogonal connection. Prove that the exponential map $\exp$ is defined on all $T_xM$, that is, for any vector $v\in T_x M$ there exists a geodesic map $\gamma:\; [0, \infty[$ tangent to $v$ \enum \ite[!] when $M$ is compact \ite[*] when it is complete as a metric space. \ite[**] Suppose that $\exp$ is defined on all $T_xM$. Prove that $M$ is complete as a metric space. \ee \ez \exercise[*] Find a compact manifold and a connection such that the exponential map is not defined on the whole $T_x M$ (that is, for some vector $v\in T_x M$ there is no geodesic map $\gamma:\; [0, 1]$ tangent to $v$). \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Jacobi fields} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $M$ be a manifold with connection, and $\exp:\; T_x M \arrow M$ the exponential map (which can be defined on the whole $T_x M$ or locally in a neighbourhood $U$ of 0). Let $A\in \End(T_x M)$ be a matrix, and $a\in T(T_x M)$ a linear vector field $x\arrow A(x)$ associated with $A$. {\bf Jacobi field} is a vector field on $\exp(U)$ obtained as an image of $a$ under the exponential map. \ed \exercise[!] Let $\gamma_0$ be a geodesic passing through $x\in M$, and $J\in TM \restrict{\gamma_0}$ a Jacobi field. Prove that there exists a smooth family $\gamma_t$ of geodesics passing through $x$ such that $J\restrict{\gamma_0(r)}= \frac{\gamma_t(r)}{dt}$. \ez \hint A flow of diffeomorphisms associated with a Jacobi field maps a geodesic passing through x to another geodesic passing through x. \eh \remark Usually, one defines Jacobi field as a vector field $J\in TM \restrict{\gamma_0}$ on a geodesic such that $J\restrict{\gamma_0(r)}= \frac{\gamma_t(r)}{dt}$ for some smooth family $\gamma_t$ of geodesics. As shown above, this definition is equivalent to ours. \er \definition {\bf Radial vector field} on a vector space $V=\R^n$ is the linear vector field associated with the map $x\arrow x$. In standard coordinates it can be written as $\sum x_i \frac{d}{dx_i}$. \ed \exercise Let $X$ be a vector field defined in a star-shaped neighbourhood of 0 in $\R^n$. Prove that $X$ is a linear vector field, if \enum \ite $X$ commutes with the radial vector field $R$ \ite $[[X, R], R]=0$. \ee \ez \hint Use the Taylor series for $X$ and the following observation: for any homogeneous vector field $R_d$ of degree $d$, one has $[R, X]=dX$. \eh \exercise Let $\exp:\; T_x M \arrow M$ be the exponential map, defining a diffeomorphism in a star-shaped neighbourhood $U\subset T_xM$, and $\theta\in TM$ the image of the radial vector field under $\exp$. Prove that $J$ is a Jacobi vector field if and only if \enum \ite[!] $[J,\theta]=0$ \ite[!] $[[J, \theta], \theta]=0$. \ee \ez \remark Abusing the language, we call the vector field $\theta\in T\exp(U)$ defined above ``the radial vector field'' as well. \er %\exercise %Let $\theta\in TM$ be the radial vector field %on a manifold with connection $\nabla$. %\enum %\ite %Prove that $\nabla_\theta \theta=0$. %\ite Let $\theta'$ be a vector field %in a neighbourhood of $x\in M$ which satisfies %$\nabla_{\theta'} {\theta}=0$. Suppose that %$\theta'\restrict x=0$. Prove that %$\theta'=\const\theta$. %\ee %\ez \definition Let $\nabla:\; TM \arrow TM \otimes \Lambda^1 M$ be a connection. {\bf Torsion} of $\nabla$ is a map $T_\nabla:\; \Lambda^2 TM \arrow TM$ mapping vector fields $X, Y\in TM$ to $\nabla_X Y -\nabla_Y X-[X, Y]$. A connection is called {\bf torsion-free} if its torsion vanishes. {\bf Curvature} of $\nabla$ is a map $\Theta_\nabla:\; \Lambda^2 TM \arrow\End(TM)$ mapping vector fields $X, Y\in TM$ to $[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$ considered as a map from $TM$ to $TM$. \ed \exercise Prove that torsion and curvature are $C^\infty(M)$-linear. \ez \exercise Let $U$ be an image of a star-shaped domain $W \subset T_x M$ under the exponential map $\exp:\; T_x M \arrow M$ which is a diffeomorphism on $W$. Denote by $\theta\in TU$ the radial vector field, and let $J\in TU$ be another vector field. Assume that the connection $\nabla$ is torsion-free. \enum \ite Prove that $J$ is Jacobi if and only if $\nabla_J \theta= \nabla_\theta J$. Prove that $J$ is Jacobi if and only if \begin{equation}\label{_Jacobi_via_3rd_commu_Equation_} \nabla_{[\theta, J]} \theta= \nabla_\theta(\nabla_\theta J-\nabla_J\theta). \end{equation} \ite Prove that $\theta$ is a radial vector field if and only if $\nabla_\theta\theta=\theta$. \ite Prove that \[ -\nabla_{[\theta, J]} = \Theta_\nabla(\theta, J)- \nabla_\theta\nabla_J + \nabla_J \nabla_\theta \] and \[ -\nabla_{[\theta, J]} \theta = \Theta_\nabla(\theta, J)(\theta)- \nabla_\theta\nabla_J\theta + \nabla_J \theta. \] Use this to rewrite \eqref{_Jacobi_via_3rd_commu_Equation_} as \begin{equation}\label{_Jacobi_via_theta_Equation_} \Theta_\nabla(\theta, J)(\theta)=\nabla_\theta\nabla_\theta(J)-\nabla_J \theta. \end{equation} \ee \ez \exercise[!] Let $J\in TM \restrict \gamma$ be a vector field tangent to the geodesic. Denote by $\ddot J$ the second derivative of $J$ along $\gamma$, with $\ddot J=\nabla_{\dot \gamma}\nabla_{\dot \gamma}(J)$. Prove that $J$ is a Jacobi field if and only if $\ddot J= \Theta_\nabla(\dot\gamma, J, \dot\gamma)$ \ez \hint Let $t$ be a parametrization of a geodesic $\gamma(t)$. Prove that $\theta$ is tangent to $\gamma$ and $t\dot \gamma = \theta$ on $\gamma$, and deduce that \[ \nabla_\theta\nabla_\theta = t^2 \nabla_{\dot \gamma}\nabla_{\dot \gamma} + t\nabla_{\dot \gamma}. \] using \eqref{_Jacobi_via_theta_Equation_}. \eh \remark $\ddot J= \Theta_\nabla(\dot\gamma, J, \dot\gamma)$ is a second order ODE. \er %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Torsion and Frobenius theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $B\subset TM$ be a sub-bundle of tangent bundle, and $\Phi:\; \Lambda^2 B \arrow TM/B$ maps vector fields $b_1, b _2\in B\subset TM$ to their commutator $[b_1, b_2]\mod B$. Then $\Phi$ is called {\bf Frobenius form}. \ed \exercise Prove that the Frobenius form is $C^\infty M$-linear. \ez \exercise Let $B \subset TM$ be a sub-bundle. Prove that there exists a connection $\nabla:\; TM \arrow TM \otimes \Lambda^1 M$ such that $\nabla(B) \subset B \otimes \Lambda^1 M$. \ez \remark In this case we say that {\bf connection $\nabla$ preserves $B$}. \er \exercise Suppose that a connection $\nabla:\; TM \arrow TM \otimes \Lambda^1 M$ preserves a sub-bundle $B\subset TM$. \enum \ite Prove that $T_\nabla$ maps $\Lambda^2 B$ to $B$ if and only if its Frobenius form vanishes. \ite[!] Suppose that the Frobenius form of $B$ vanishes. Prove that there exists a torsion-free connection preserving $B\subset TM$ \ee \ez \hint Use the same argument as used in the proof of existence of Levi-Civita connection. \eh \exercise Let $B\subset TM$ be a sub-bundle, preserved by a connection $\nabla$, and $\Theta_\nabla$ its curvature. Prove that $\Theta_\nabla(X, Y, Z)\in B$ whenever $Z\in B$. \ez \exercise Let $B\subset TM$ be a sub-bundle, preserved by a torsion-free connection $\nabla$. \enum \ite Denote by $\theta$ the radial vector field, defined in a neighbourhood of $x\in M$. Prove that for any Jacobi field $J$ such that $J\restrict x \in B$ and $\nabla_\theta J\restrict x \in B$, one has $J\in B$. \ite[!] Use his to show that the exponential map maps the subspace $B\restrict x \subset T_x M$ to a submanifold of $M$ tangent to $B$. \ee \ez \exercise[!] Prove Frobenius theorem: given $B\subset TM$ such that $[B, B]\subset B$, for any $x\in M$ there exists a neighbourhood $U \subset M$ and a family of submanifolds $V_t\subset U$ parametrized by $t\in Z$ such that $U$ is obtained as a disjoint union of $V_t$, and $TV_t = B$ at each point of $V_t$ for each $t\in Z$. \ez \remark In these assumptions, submanifolds $V_t$ are called {\bf leaves of the foliation} defined by $B$, and $B$ is called {\bf an involutive} or {\bf holonomic} sub-bundle of $TM$. \er \end{document}