\documentclass[11pt]{article} \usepackage{url} %version 1.0,\ \ 26.02.2018, partially translated from hodge4.tex % then moved most of it to 7 % version 1.1, 28.02 Katya Amerik prislala ispravlenij % version 1.2, 17.03.2018: small changes, 19b became (**) \newcommand{\version}{version 1.2,\ \ 17.03.2018} \newcommand{\firstdate}{28.02.2018} \addtolength{\topmargin}{-10mm} \addtolength{\textheight}{20mm} \addtolength{\oddsidemargin}{-7mm} \addtolength{\textwidth}{15mm} \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{6}{Hodge theory 6: Connections} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\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{Connections} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $B$ be a vector bundle on a smooth manifold $M$, and \[ \nabla:\; B \arrow B \otimes \Lambda^1 M \] a differential operator which satisfies \[ \nabla(fb) = b \otimes df + f \nabla b, \] for any $f\in C^\infty(M)$ and any $b\in B$. Then $\nabla$ is called {\bf connection} on $B$. Given a vector field $X$, consider an operator $\nabla_X:\; B \arrow B$ obtained by the convolution of $\nabla(b)$ with $X$. This operator is called {\bf covariant derivative} along $X$. \ed \exercise Prove that the covariant derivative $\nabla_X$ satisfies the Leibniz rule $\nabla_X(fb)= \langle df, X\rangle b + f \nabla_X b$. Here $\langle df, X= \Lie_X f = df\cntrct X$ denotes the derivative of $f$ along $X$. \ez \exercise Let $B$ be a vector bundle on $M$. Suppose that for any vector field $X\in TM$ we are provided with a covariant derivative operator $\nabla_X:\; B \arrow B$ satisfying the Leibniz rule and $C^\infty M$-linear on $X$. Prove that $\nabla_X$ is obtained from a connection. \ez \exercise Prove that the symbol $\Symb(\nabla)\in TM \otimes \Hom(B, \Lambda^1 M \otimes B)$ of a connection is given by an identity map $\Id_{TM}\otimes \Id_B:\; TM \otimes T^* M \otimes \Hom(B,B)$. \ez \exercise[!] Let $D$ be a first order differential operator on a bundle $B$ with symbol $\Symb(D)\in TM \otimes \Hom(B, \Lambda^1 M \otimes B)$ given by an identity map $\Id_{TM}\otimes \Id_B:\; TM \otimes T^* M \otimes \Hom(B,B)$. Prove that it is a connection. \ez \exercise[!] Let $\nabla$ be a connection on $B$, $B^*$ the dual bundle. Prove that there exists a unique operator $\nabla^*:\; B^* \arrow B^* \otimes \Lambda^1 M$ such that \[ d\langle b , b'\rangle = \langle \nabla b, b'\rangle + \langle b, \nabla^* b'\rangle \] for any $b\in B, b' \in B^*$. Prove that $\nabla^*$ is a connection on $B^*$. \ez \exercise Let $B_1, ..., B_n$ be vector bundles with connections, denoted by $\nabla$ (people often use the same letter $\nabla$ to denote different connections if they are defined on different bundles). Consider the following differential operator \[ \nabla:\; B_1 \otimes ... \otimes B_n \arrow B_1 \otimes ... \otimes B_n \otimes \Lambda^1 M, \] $\nabla(b_1 \otimes b_2 \otimes ... \otimes b_n)= \nabla(b_1) \otimes b_2 \otimes ... \otimes b_n + b_1 \otimes \nabla(b_2) \otimes ... \otimes b_n + ... + b_1 \otimes b_2 \otimes ... \otimes \nabla(b_n)$. Prove that $\nabla$ defines a connection on the vector bundle $B_1 \otimes ... \otimes B_n$. \ez \remark Previous two exercises show that a connection on a bundle $B$ defines a connection on any tensor power $B^{\otimes n} \otimes (B^*)^{\otimes m}$. This connection is almost always denoted by the same letter. \er \exercise[!] Let $B$ be a vector bundle over a manifold admitting partition of unity. Prove that $B$ admits a connection. \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Holonomy} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $(B, \nabla)$ be a bundle with connection. A tensor $\Psi\in B^{\otimes i} \otimes (B^*)^{\otimes j}$ is called {\bf parallel} if $\nabla(\Psi)=0$. In this case we also say that $\Psi$ {\bf is preserved} by $\nabla$. \ed \exercise Let $g$ be a tensor on a bundle $B$ over $M$. Construct a connection $\nabla$ such that $\nabla(g)=0$ if \enum \ite[!] $g$ is a non-degenerate bilinear symmetric form on $B$. \ite[*] $g$ is a non-degenerate bilinear antisymmetric form on $B$. \ite[*] $g$ is a bilinear symmetric form of constant rank on $B$. \ee \ez \exercise Let $B$ be a trivial vector bundle with connection over $\R$. Prove that for each $x\in \R$ and each vector $b_x \in B\restrict x$ there exists a unique section $b\in B$ such that $\nabla b=0$, $b\restrict x= b_x$. \ez \definition Let $\gamma:\; [0, 1] \arrow M$ be a smooth path in $M$ connecting $x$ and $y$, and $(B, \nabla)$ a vector bundle with connection. Restricting $(B, \nabla)$ to $\gamma[0,1]$, we obtain a bundle with connection on an interval. Solve an equation $\nabla(b)=0$ for $b\in B\restrict{\gamma([0,1])}$ and initial condition $b\restrict x= b_x$. This process is called {\bf parallel transport} along the path via the connection. The vector $b_y:= b\restrict y$ is called {\bf vector obtained by parallel transport of $b_x$ along $\gamma$}. {\bf Holonomy group} of $\gamma$ is the group of endomorphisms of the fiber $B_x$ obtained from parallel transports along all paths starting and ending in $x\in M$ \ed \exercise Let $(B, \nabla)$ be a vector bundle over a connected manifold $M$, and $x, y \in M$. Construct an isomorphism of the corresponding holonomy groups $\Hol_x(\nabla)\arrow \Hol_y(\nabla)$. \ez \exercise Find a bundle with connection over $S^1$ which has non-trivial holonomy. \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Iterated connection} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $M$ be a smooth manifold. A connection on $TM$ or on $\Lambda^1M$ is called {\bf connection on $M$}. This connection defines a connection on all tensor powers of $TM$ and $\Lambda^1 M$. A tensor product of several copies of $TM$ and $\Lambda^1 M$ is called {\bf a tensor bundle} on $M$, and its section {\bf a tensor}. Similarly, a section of a tensor product of several copies of $B$ and $B^*$ is called {\bf a tensor over a bundle $B$}. \ed \definition Let $B$ be a vector bundle with connection $\nabla_0$ over a manifold $M$, and $\nabla$ a connection on $\Lambda^1 M$. Define a connection \begin{equation}\label{_nabla_i_Equation_} \nabla_i:\; B\otimes \underbrace{\Lambda^1M\otimes ...}_\text{{$i$ times}}\arrow B\otimes \underbrace{\Lambda^1M\otimes ...}_\text{{$i+1$ times}} \end{equation} using the Leibniz formula \[ \nabla_i(b \otimes \xi_1 \otimes ... \otimes \xi_i) = \nabla_{i-1}(b \otimes \xi_1 \otimes ... \otimes \xi_{i-1}) \otimes \xi_i+ b \otimes \xi_1 \otimes ... \otimes \xi_{i-1} \otimes \nabla \xi_i. \] Denote by \[ \nabla^i:\; B \arrow B\otimes \underbrace{\Lambda^1M\otimes ...}_\text{$i$ times} \] the composition $\nabla_0 \circ \nabla_1 \circ ... \circ \nabla_{i}$. This operator is called {\bf an $i$-th power of the connection $\nabla$}. \ed \exercise \enum \ite Prove that the symbol of $\nabla^2$, considered as an element of \[ \Sym^2 TM \otimes \Hom (B, B\otimes \Lambda^1M\otimes\Lambda^1M) \] is symmetric under the permutation of the tensor multipliers $\Lambda^1M\otimes\Lambda^1M$. \ite[*] Let $S$ be the symbol of $\nabla^i$, \[ S\in \Sym^i TM \otimes \Hom \bigg (B, B\otimes \underbrace{\Lambda^1M\otimes ...}_\text{$i$ times}\bigg) \] Prove that $S$ is symmetric under the permutations of the tensor multipliers $\Lambda^1M\otimes\Lambda^1M\otimes ... \otimes \Lambda^1 M$. \ee \ez \exercise\label{_diff_op_via_conne_Exercise_} Let $D\in \Diff^s(B,C)$ be a differential operator on vector bundles $B$, $C$. \enum \ite[!] Prove that there exists a $C^\infty$-linear map \[ \Psi:\; B \otimes \bigoplus_{i=0}^s(\Lambda^1 M)^{\otimes i}\arrow C\] such that $D(b)=\Psi\left(\bigoplus_{i=0}^s \nabla^i b\right)$. \ite[**] Prove that there exists a $C^\infty$-linear map and a connection $\nabla$ such that $\Phi:\; B \otimes (\Lambda^1 M)^{\otimes s}\arrow C$ such that $D(b)=\Phi(\nabla^s b)$. \ee \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Adjoint operators} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \exercise[!]\label{_first_order_via_nabla_Exercise_} Let $D:\; B \arrow C$ be a first order differential operator on a vector bundle with connection $\nabla$. Prove that $D$ is obtained as a linear combination of $b \arrow A(\nabla_{X_i})$, where $X_i$ are vector fields and $A:\; B \arrow C$ is a linear map. \ez \definition Let $\Vol$ be a volume form on a manifold $M$. {\bf Divergence} of a vector field $X$ is the form $\Lie_X \Vol = d(\Vol\cntrct X)$. \ed \exercise[!]\label{_div_free_generate_Exercise_} Let $\Vol$ be a volume form on a manifold $M$, and ${\cal D}\subset TM$ the sheaf of vector fields with zero divergence. Prove that the global sections of the sheaf ${\cal D}$ generate all vector fields on over $C^\infty M$. \ez \exercise Let $B$ be a vector bundle over a manifold $M$, $\Vol$ a volume form and $\nabla$ a connection on $B$. Consider the pairing between sections of $B$ and the dual bundle $B^*$, with \begin{equation}\label{_section_pairing_via_Vol_Equation_} \langle b, \lambda\rangle = \int_M (b, \lambda) \Vol \end{equation} for each $b\in B, \lambda\in B^*$. \enum \ite Let $X\in TM$ a vector field with zero divergence. Prove that the operators $\nabla_X:\; B \arrow B$ and $\nabla_X^*:\; B^* \arrow B^*$ are adjoint with respect to the pairing \eqref{_section_pairing_via_Vol_Equation_}. \ite Let $X$ be an arbitrary vector field, and $\nabla_X:\; B \arrow B$ the corresponding covariant derivative. Consider the adjoint $(\nabla_X)^*:\; B^* \arrow B^*$ associated with the pairing \eqref{_section_pairing_via_Vol_Equation_}. Prove that \[ \int_M (b, \nabla_X^* b')\Vol + \int_M (\nabla_X b, b')\Vol = -\int_M (b,b') \Lie_X\Vol \] \ite[!] Prove that $(\nabla_X)^*$ is a differential operator. \ee \ez \exercise Prove that an adjoint map of a first order differential operator on vector bundles $D:\; B \arrow B_1$ is a first order differential operator from $B_1^*$ to $B^*$. \ez \hint Use Exercise \ref{_first_order_via_nabla_Exercise_}. \eh \end{document}