\tableofcontents

\chapter{Tosion Invariants [Roman Sauer]}

Torsion invariants fall into a class of so-called ``secondary invariants'' of topological spaces in the sense that they are only defined if a certain class of ``primary invariants'' (e.g. Betti numbers) vanish.
Often they reveal more subtle geometric information.
The following will contain a discussion of Whitehead and Reidemeister torsion.
Informally, corresponding primary invariants are Lefschetz numbers (Whitehead torsion) and the Euler characteristic (Reidemeister torsion).

\section{Review of Euler characteristic and Lefschetz numbers.}

\subsection{CW Complexes}

\begin{dfn*}
  A (finite) \CmMark{CW-complex} is a hausdorff space with a decomposition $E$ into (finitely many) cells (space hemeomorphic to some $\R^n$) such that for every $e \in E$ there is a continuous map $\phi_e \colon D^n \to X$ with $\phi_e \colon \mathring D^n \xrightarrow{\cong} e$ and $\Im(\phi_e|_{S^{n-1}}) \subset \bigcup_{f \in E, \dim f \leq n-1} f$.
\end{dfn*}

\begin{expl*}
  \begin{enumerate}
  \item Simplicial complexes, e.g. triangles, pyramides, etc.
  \item But CW-complexes are more general, the following graph is CW for example:
    \begin{center}
      \begin{tikzpicture}
        \draw (0,0) to[bend left] (2,0);
        \draw (0,0) to[bend right] (2,0);
        \draw (2,0) to (3,0);
      \end{tikzpicture}
    \end{center}
    One can even attach a disc along its boundary to a single 1-cell.
  \end{enumerate}
\end{expl*}

\subsection{Euler characteristic}

\begin{dfn*}
  The Euler class $\chi(X)$ of a finite CW-complex $X$ is defined as $\chi(X) = \sum_{i \geq 0}(-1)^i \#(i\text{-cells of } X) \in \Z$.
\end{dfn*}

\begin{thm*}[Euler-Poincaré]
  \begin{align*}
    \chi(X) = \sum_{i \geq 0} (-1)^i b_i(X),
  \end{align*}
  where $b_i(X) = \rk_{\Z} H_i(X;\Z)$.
\end{thm*}

In particular, $\chi$ is a homotopy invariant.

\begin{proof}[``Proof'']
  $H_i(X;\Z) = H_i(C_{*}^{CW}(X))$, where $C_{*}^{CW}(X)$ is the cellular chain complex
  \begin{align*}
    \cdot \to C_{i+1}^{CW}(X)\xrightarrow{\partial} \underbrace{C_{i}^{CW}(X)}_{\cong \Z^{\# i\text{-cells}}} \xrightarrow{\partial} C_{i-1}^{CW}(X) \to \cdots
  \end{align*}
  Thus $\chi(C_{*}) := \sum_{i \geq 0} (-1)^i\rk_{\Z}(C_{i})$ and $\chi(C^{CW}(X)) = \chi(X)$.
  This boils down to
  \begin{align*}
    \chi(C_{*}) = \sum_{i \geq 0} \rk_{\Z}H_i(C_{*}) ( = \chi(H_{*}(C_{*}))].
  \end{align*}
  This is just additivity of the rank!
  Consider
  \begin{align*}
    C_1 \xrightarrow{\partial} C_0
  \end{align*}
  and note that we have the exact sequences $0 \to \Im \partial \to C_0 \to \underbrace{H_0}_{= C_0/\Im \partial} \to 0$ and $0 \to \underbrace{H_1}_{= \Ker \partial} \to C_1 \xrightarrow{\partial} \Im \partial \to 0$.

  Thus $\chi(C_{*)} = \rk_{\Z} C_0 - \rk_{\Z} C_1 = \rk_{\Z} \Im \partial + \rk_{\Z} H_0 - \rk_{\Z}H_1 - \rk_{\Z} \Im \partial = \rk H_0 - \rk H_1$, which completes the ``proof''.
\end{proof}

\subsection{Review of cellular homology}

Let $X$ be a CW-complex with cellular decomposition $E$.
Then we can consider the \CmMark{n-skeleton}
\begin{align*}
  X^n := \sum_{e \in E, \dim e \leq n} e,
\end{align*}
which yields a filtration $X^0 \subset X^1 \subset \cdots \subset X$ such there is a pushout diagram
\begin{equation*}
  \begin{tikzcd}
    \coprod S^{n-1} \ar{r} \ar[hook]{d} & X^{n-1} \ar[hook]{d} \\
    \coprod D^n \ar{r} & X^n
  \end{tikzcd}
\end{equation*}
One could take this as an alternative definition of a CW-complex by a filtration with the pushout property.
The cells can be recovered as connected components of $X^n\setminus X^{n-1}$.

We have
\begin{align*}
  C_i^{CW}(X) = H_i(X^i, X^{i+1}) \xleftarrow{\cong} H_i(\coprod D^i, \coprod S^{i-1}) \cong \bigoplus H_i(D^i, S^{i-1}) \cong \bigoplus \Z^{\# i\text{-cells}},
\end{align*}
where the first isomorphism $\leftarrow$ is given by excision.
The boundary maps $C_i^{CW}(X) \xrightarrow{\partial} C_{i-1}^{CW}(X) $ come from
\begin{align*}
  H_i(X^i,X^{i-1}) \to H_{i-1}(X^{i-1}) \to H_{i-1}(X^{i-1},X^{i-2}).
\end{align*}
Under this isomorphism, the matrix entry belonging to $(e,f)$ where $e$ is an $n$-cell, $f$ an $(n-1)$-cell is the \CmMark{degree} of the map.
\begin{align*}
  S^{i-1} \xrightarrow{\phi_e|_{S^{n-1}}} X^{i-1} \xrightarrow{\operatorname{proj}} X^{i-1}/(X^{i-1}\setminus f) \xleftarrow{\phi_f, \cong} D^{i-1}/S^{i-2} \cong S^{i-1}.
\end{align*}

\begin{expl*}
  Consider the torus as an identification square.
  We convince ourselves that the cellular chain complex is given as $\Z \to \Z \oplus \Z \to \Z$, where $1 \mapsto (0, 0)$, since it is described by a map $S^1 \to S^1$ traversing the 2-cell according to orientation has degree $0$.
\end{expl*}


\subsection{Lefschetz number}

Recall that a map $f \colon X \to P$ between CW-complexes is \CmMark{cellular}, if $f(X^i) \subset Y^i$ for all $i$.

\begin{thm*}[Cellular approximation]
  Any map between CW-complexes is homotopic to a cellular map.
\end{thm*}

\begin{dfn*}
  The \CmMark{Lefschetz number} of a self-map $f \colon X \to X$ of a finite CW-complex is defnined as
  \begin{align*}
    \Lambda(f) = \sum_{i \geq 0} (-1)^i\tr C_i^{CW}(f) \in \Z.
  \end{align*}
\end{dfn*}

\begin{rem*}
  $\Lambda(\id_X) = \chi(X)$.
\end{rem*}

The following theorem yields a description of Lefschetz numbers by homology.
\begin{thm*}
  $\Lambda(f) = \sum_{i \geq 0}(-1)^i \tr H_i(f)$.
\end{thm*}
Thus, this number only depends on the homotopy class of $f$.

\begin{proof}
  Similar to the proof of Euler-Poincaré using the additivity of the trace, i.e. in the situation
  \begin{equation}
    \begin{tikzcd}[row sep=small]
      0 \ar{r} & A \ar{r} \ar{d}{a} & B \ar{r} \ar{d}{b} & C \ar{r} \ar{d}{c} & 0\\ 
      0 \ar{r} & A \ar{r} & B \ar{r} & C \ar{r} & 0
    \end{tikzcd}
  \end{equation}
  we have $\tr(b) = \tr(a) + \tr(c)$.
\end{proof}

\begin{thm*}
  If $f$ has no fixed point, then $\Lambda(f) = 0$.
\end{thm*}

\begin{rem*}
  The converse is not true (think of counterexamples, e.g. $S^1 \wedge S^1$), although there is one in the case of simply-connected closed manifolds.
\end{rem*}

\begin{proof}
  Let $X$ be metrizable and let $d$ be a metric.
  If $X$ is compact, there exists an $\varepsilon > 0$ with $d(f(x), x) > 3\varepsilon$.
  One can ``refine'' the CW-structure to a new one such that every cell has diameter $< \varepsilon$.
  By cellular approximation we can see that there exists a cellular map $g \colon X \to X$ with $g \simeq f$ and $d(g(x),f(x)) < \varepsilon$.
  Thus $g(\overline e) \cap \overline e = \emptyset$ for every cell $e$.
  Hence, the diagonal matrix entries of each $C_i^{CW}(g)$ are zero and thus $\Lambda(g) = \Lambda(f) = 0$.
\end{proof}


\section{Whitehead torsion}

\subsection{Introduction/Motivation}

Given a homotopy equivalence $f \colon X \xrightarrow{\simeq} Y$ of finite CW-complexes, Whitehead torsion is an assignment $\tau(f) \in \Wh(\pi_1(Y))$ living in the so-called Whitehead group.

\begin{thm*}[Properties of Whitehead torsion]
  \begin{enumerate}[label=(\arabic*)]
  \item homotopy invariance
  \item\footnote{This is a deep theorem of Chapman.} If $f \colon X \to Y$ is a homeomorphism, then $\tau(f) = 0$.
  \item additivity:
    A cellular pushout is a diagram
    \begin{equation*}
      \begin{tikzcd}
        X_0 \ar{r}{f} \ar[hook]{d}{i} & X_2 \ar{d}\\
        X_1 \ar{r} & X
      \end{tikzcd}
    \end{equation*}
    with $X_i$ be CW-complexes, where $f$ is cellular and $i$ is an inclusion of a subcomplex.
    If the diagram
        \begin{equation*}
      \begin{tikzcd}
        X_0 \ar{rr} \ar[hook]{dd} \ar{rd}{f_0}[swap]{\simeq} &[0.8cm] &[-0.2cm] X_2 \ar[bend left=10]{rd}{f_2}[swap]{\simeq} &[0.8cm] \\
        & Y_0 \ar[near start]{rr}{\phi} \ar[near end,hook]{dd}{j} & & Y_2 \ar{dd}{i}\\
        X_1 \ar[crossing over]{rr} \ar[bend right=10]{rd}{f_1}[swap]{\simeq} & & X \ar[dashed]{rd}{f} \ar[leftarrow,crossing over]{uu} & \\
        & Y_1 \ar{rr}{\psi} & & Y
      \end{tikzcd}
    \end{equation*}
    is a map of cellular pshouts such that $f_i$ are homotopy equivalences.
    Then $f$ is a homotopy equivalence and
    \begin{align*}
      \tau(f) & = "\tau(f_1) + \tau(f_2) - \tau(f_0)"\\
      & = \psi_{*}(\tau(f_1))  + j_{*}(\tau(f_2)) - (\psi \circ i)_{*}(\tau(f_0)) \in \Wh(\pi_1(Y)).
    \end{align*}
   A similar additivity holds for the Lefschetz number.\footnote{Idea. $0 \to C_1(X_0) \to C_{*}(X_1) \oplus C_{*}(X_2) \to C_{*}(X) \to 0$ exact.}
 \item composition formula.
   If we have
   $\begin{tikzcd}
     X \ar{r}{f}[swap]{\simeq} & Y \ar{r}{g}[swap]{\simeq} & Z
   \end{tikzcd}$,
   then
   \begin{align*}
     \tau(g \circ f) & = "\tau(f) + \tau(g)"\\
     & = g_{*}(\tau(f) + \tau(g)) \in \Wh(\pi_1Z).
   \end{align*}
  \end{enumerate}
\end{thm*}

\begin{thm*}[s-cobordism theorem (Mazur, Barden, Stallings, Smale)]
  Let $M$ be a closed smooth manifold of dimension $\geq 5$.
  Let $(W, i, j)$ be an s-cobordism
  \begin{figure}[h!]
    \centering
    \begin{tikzpicture}[scale=0.8]
      
      \coordinate (L1) at (2,0);
      \coordinate (L2) at (2,3);

      \coordinate (LL1) at ($(L1)+(-4,0)$);
      \coordinate (LL2) at ($(L2)+(-4,0)$);
      
      \coordinate (R1) at (8,0.5);
      \coordinate (R2) at (8,2);

      \coordinate (RR1) at ($(R1)+(3,0)$);
      \coordinate (RR2) at ($(R2)+(3,0)$);

      % left inclusion
      
      \ellipsebetweenvert{LL1}{LL2}

      \node[below] at (LL1) {$M$};
     
      \draw[right hook->] ($(LL1) + (1.2,1.5)$) to node[above] {$i$} node[below]{$\simeq$} ($(L1) + (-1.2,1.5)$);

      % cobordism

      \ellipsebetweenvert{L1}{L2}
      \node[below] at (L1) {$M_0$};

      \draw[out=0,in=180] (L1) to (5,-1) to (R1);
      \draw[out=0,in=180] (L2) to (R2);

      \topgenus[0.35]{5,0}
      \topgenus[0.22]{6,1.2}

      \node at (6,3) {$W$};
      
      \ellipsebetweenvert[left]{R1}{R2}
      \node[below] at (R1) {$M_1$};

      % right inclusion

      \draw[left hook->] ($(RR1) + (-1,0.75)$) to node[above] {$j$} node[below]{$\simeq$} ($(R1) + (1,0.75)$);

      \ellipsebetweenvert{RR1}{RR2}
      
      \node[below] at (RR1) {$N$};
    \end{tikzpicture}
    % sketch: \includegraphics[width=0.8\textwidth]{img/1.png}
  \end{figure}
  
  i.e. $\partial W = M_0 \coprod M_1$ and $i \colon M \hookrightarrow W$, $j \colon N \hookrightarrow W$ are homotopy equivalences.
  Then $\tau(M \xrightarrow{i} W) = 0$ if and only if $(W,i_0,i_1)$ is trivial, i.e.
  \begin{figure}[h!]
    \centering
    \begin{tikzpicture}[every node/.style={scale=0.8}]
      \coordinate (LU1) at (2.4,1.5);
      \coordinate (LU2) at (2.4,2.5);
      
      \coordinate (RU1) at (5,1.3);
      \coordinate (RU2) at (5,2.3);

      \coordinate (LD1) at (2.4,0.5);
      \coordinate (LD2) at (2.4,-0.5);
      
      \coordinate (RD1) at (5,0.5);
      \coordinate (RD2) at (5,-0.5);

      % leftmost part
      
      \node at (0,3) {$\exists$};

      \node[scale=1.5] at (-1,1) {$M$};

      \draw[right hook->] (-0.4,1.2) to node[above] {$i$} (2,2);
      \draw[right hook->] (-0.4,0.8) to node[below] {$\operatorname{incl}$} (2,0);

      % upper cobordism

      \ellipsebetweenvert{LU1}{LU2}

      \draw[out=0,in=180] (LU1) to (3,1.6) to (RU1);
      \draw[out=0,in=180] (LU2) to (4,2.2) to (RU2);

      \ellipsebetweenvert[left]{RU1}{RU2}

      % arrow from the cobordism to the cylinder

      \draw[->] ($(LU1)!0.5!(RU1) + (0,-0.1)$) to node[left]{$\cong$} node[right]{diffeo} ($(LU1)!0.5!(RU1) + (0,-0.7)$);

      % cylinder

      \ellipsebetweenvert{LD1}{LD2}

      \draw (LD1) to (RD1);
      \draw (LD2) to (RD2);
      
      \ellipsebetweenvert[left]{RD1}{RD2}
    \end{tikzpicture}
    % sketch: \includegraphics[width=0.5\textwidth]{img/2.png}
  \end{figure}
\end{thm*}

This theorem implies the Poincaré conjecture in dimensions at least 6, which says that if $M$ is a closed (smooth) manifold that is homotopy equivalent to $S^n$, then $M$ is homeomorphic to $S^n$.

The proof of this implication is basically along these lines:
Pick disjoint embedded $n$-disks in $M$ and remove them.
The result is a manifold $W$ with two boundary components.
Consider the s-cobordism theorem for this manifold as depicted in the figure below, where $f$ is a diffeomorphism of $(n-1)$-spheres.
\begin{figure}[h!]
  \centering
  \begin{tikzpicture}
    \coordinate (U1) at (0,3);
    \coordinate (U2) at (2.2,3);
    
    \coordinate (D1) at (-0.2,0);
    \coordinate (D2) at (1.8,0);

    % bordism on the left
    
    \ellipsebetweenhor{U1}{U2}

    \draw[out=270,in=90] (U1) to (0.3,1.8) to (D1);
    \draw[out=270,in=90] (U2) to (2.3,1.7) to (D2);
    
    \ellipsebetweenhor[upper]{D1}{D2}

    \node[left] at (0,1.5) {$W$};

    % arrows and lower disk

    \draw[bend left=15,->] (2.8,3.3) to node[above] {$f$} (4.2,3.3);

    \draw[->] (3,1.5) to node[above] {$\cong$} (4,1.5);

    \draw[left hook->] (2.5,-1) to (1.3,-0.5);
    \draw[right hook->] (4.2,-1) to (5.5,-0.5);

    \draw[pattern=north west lines,pattern color=gray] (3.35,-1) ellipse (0.6 and 0.3);
    \node at (3.8,-0.4) {$D^n$};

    % cylinder on the right

    \ellipsebetweenhor{5,3}{7,3}

    \draw (5,3) to (5,0);
    \draw (7,3) to (7,0);
    
    \ellipsebetweenhor[upper]{5,0}{7,0}
    
    \node[right] at (7,1.5) {$S^{n-1} \times [0,1]$};
      
  \end{tikzpicture}
  % sketch: \includegraphics[width=0.7\textwidth]{img/3.png}
\end{figure}

By filling top and bottom, the Poincaré conjecture is implied, if we can extend a diffeomorphism $f \colon S^{n-1} \xrightarrow{\cong} S^{n-1}$ to a homeomorphism $F \colon D^n \xrightarrow{\cong} D^n$.
This can be done by the so-called Alexander trick ($F(t x) = tf(x)$ for $t \in [0,1], x\in S^{n-1}$).


\subsection{Whitehead group and lower K-theory}

Let $R$ be a unital ring.
Then
\begin{align*}
  K_0(R) := \left< G \mid R \right>_{\text{ab}}
\end{align*}
with generators $G = \{$ isomorphism classes $[P]$ of fin. gen. projective $R$-modules $\}$ and relations $R = \{\ [P_1] = [P_0] + [P_2]$ whenever $0 \to P_0 \to P_1 \to P_2 \to 0$ is exact $\}$.
(Recall that a direct summand of in a free $R$-module is called a \CmMark{projective module}.)
This can be understood as a kind of universal dimension for projective $R$-modules.
\begin{align*}
  K_1(R) := \left< G \mid R \right>_{\text{ab}}
\end{align*}
with generators $G = \{$ conjugacy classes $[f]$ of automorphisms $f \colon P \to P$ of fin. gen. projective $R$-modules $\}$ and relations $R$ given as follows.

\begin{enumerate}[label=(\roman*)]
\item Every commuting diagram 
  \begin{equation*}
  \begin{tikzcd}
    0 \ar{r} & P_0 \ar{r} \ar{d}{f_0}[swap]{\cong} & P_1 \ar{r} \ar{d}{f_1}[swap]{\cong} & P_2 \ar{r} \ar{d}{f_2}[swap]{\cong} & 0\\
    0 \ar{r} & P_0 \ar{r} & P_1 \ar{r} & P_2 \ar{r} & 0.
  \end{tikzcd}
\end{equation*}
gives rise to a relation $[f_1] = [f_0] + [f_2]$.
\item $f,g \colon P \xrightarrow{\cong} P$ yield a relation $[f \circ g] = [f] + [g]$.
\end{enumerate}

This can be understood as an attempt to define a universal determinant of an automorphism.

There is a more common definition of $K_1(R)$ in terms of the general linear groups with coefficients in $R$.
Recall that $\GL(R) = \colim_{n \to \infty} \GL_n(R)$ with respect to the inclusion $\GL_n(R) \hookrightarrow \GL_{n+1}(R)$ to the upper left block.
Then
\begin{align*}
  K_1(R) = \GL(R)_{\text{ab}} = \GL(R)/[\GL(R),\GL(R)].
\end{align*}

The so-called \CmMark{Whitehead lemma} states that $[\GL(R),\GL(R)] = E(R)$, where $E(R)$ is the subgroup of $\GL(R)$ generated by all elementary upper triangular matrices with ones on the diagonal.
As a consequence, if $R$ is a field then the determinant defines an isomorphism $\det \colon K_1(R) \xrightarrow{\cong} \R\setminus\{0\}$.

To see the equivalence of these two definitions, we can use the map
\begin{align*}
  \GL(R) & \to K_1(R)\\
  A & \mapsto [R^n \to R^n, \ x \mapsto Ax]
\end{align*}
and the fact that is descents to $\GL(R)_{\text{ab}} \to K_1(R)$.

The inverse homomorphism is given by $R^n \cong \{ P \oplus Q \xrightarrow{f \otimes \id} P \oplus Q \mid f \text{ iso } \}$.

\begin{dfn*}
  Let $\Gamma$ be a group.
  Then the \CmMark{Whitehead group} is defined as
  \begin{align*}
    \Wh(\Gamma) := \coker(\Gamma \times \{\pm 1\} \to K_1(\Z[\Gamma]), \ (\gamma, \pm 1) \mapsto \pm[\gamma])
  \end{align*}
\end{dfn*}

\begin{expl*}
  The Whitehead group $\Wh(\{1\})$ is trivial, since the determinant yields an isomorphism $K_1(\Z) \xrightarrow{\det} \{\pm 1\}$.

  It is a conjecture that torsion-free groups $\Gamma$ have vanishing Whitehead group.

  Assertion: $\Wh(\Z/5) \cong \Z$.
  Here only prove that $\Wh(\Z/5)$ is infinite.
  We have a map $\phi_{*} \colon K_1(\Z[\Z/5]) \to K_1(\C)$ induced by $\Z[\Z/5] \xrightarrow{\phi} \C, \ t \mapsto \xi$, where $\Z/5 \cong \left<t\right>$ and $\xi = \exp(2\pi i/5) \in \C$.
  Thus
  \begin{equation*}
    \begin{tikzcd}
      K_1(\Z[\Z/5]) \ar{r}{\phi_{*}} \ar{d} & K_1(\C) \ar{r}{\det} & \C^{\times} \ar{r}{|\blank|} & \R_{> 0}\\
      \Wh(\Z/5) \ar[bend right=10]{rrru}{\tau} & & &
    \end{tikzcd}
  \end{equation*}
  One can see that $1 - t - t^{-1}$ is a unit in $\Z[\Z/5]$, since $(1 - t - t^{-1})( - t^2 - t^3) = 1$ and thus $\tau([1 - t - t^{-1}]) \neq 1$
\end{expl*}


\chapter{Harmonic Maps [Andy Sanders]}

Also consider the notes \url{www.mathi.uni-heidelberg.de/~asanders/harmonicmaps.htm}.

\section{Basics of harmonic maps}

In the following let every manifold be oriented (for integration safety reasons).

\subsection{Background differential geometry}

Let $E \to M$ be an $\R$-vector bundle over $M$ (second countable, hausdorff manifold) of rank $r$.
A \CmMark{connection} $\nabla$ on $E$ is an $\R$-linear map
\begin{align*}
  \nabla \colon \Omega^0(E) \to \Omega^0(\T^{*}M \otimes_{\R} E) =: \Omega^1(M,E),
  s \mapsto \nabla_{\blank} s
\end{align*}
where $\Omega^0(E)$ denotes smooth sections in $E$, such that
\begin{enumerate}
\item $\nabla_{X+Y}s = \nabla_Xs + \nabla_Ys$,
\item $\nabla_X(s+s') = \nabla_X s + \nabla_Xs'$
\item $\nabla_{fX} s = f\nabla_Xs$
\item $\nabla_X(fs) = f\nabla_Xs + X(f) s$.
\end{enumerate}

Let $q$ be an inner product on $E$.
We say that $\nabla$ is a \CmMark{metric connection} for $q$, if for all $s,t \in \Omega^0(E)$ we have
\begin{align*}
  \dop q(s,t) = q(\nabla s,t) + q(s, \nabla t).
\end{align*}

\begin{expl*}
  Let $(M,g)$ be a riemannian manifold with tangent bundle $E = \T M$ and Levi-Civita connection $\nabla$ of $g$.

  Let $X,Y \in \Omega^0(M)$ be vector fields, i.e. $X = X^i \frac{\partial}{\partial x^i}$ and $Y = Y^j \frac{\partial}{\partial x^j}$ in local co-ordinates.
  (Abbreviate $\partial_i$ for $\frac{\partial}{\partial x^i}$.)
  \begin{align*}
    \nabla_XY = \nabla_{X^i\partial_i} Y^i\partial_i
    = X^i(\nabla_{\partial_i}Y^i\partial_i)
    = X^i(\partial_iY^i\partial_i + Y^i\nabla_{\partial_i}\partial_i)
    = X^i(\partial_iY^i\partial_i + Y^i\Gamma_{ij}^k\partial_i)
  \end{align*}
  where $\Gamma_{ij}^k = g^{km}(\partial_ig_{im} + \partial_j g_{im} - \partial_mg_{ij})$ for $g_{ij} = g(\partial_i,\partial_j)$ and $g^{km}$ is the $km$-entry of $g^{-1}$.
\end{expl*}

Out of $E$ one can build another bundle $E^{*} = \Hom(E,\R)$ and given another vector bundle $F$, one can build $\Hom(E,F)$, 

\begin{dfn*}
  Let $(E,\nabla) \to M$ be a vector bundle with a connection over $M$.
  The space of \CmMark{$p$-forms} on $m$ with values in $E$ is the $C^{\infty}(M)$-module $\Omega^p(M,E) = \Omega^0(M,\bigwedge^p\T^{*}M \otimes E)$.
  Elements $\alpha$ in $\Omega^p(M,E)$ have representations as linear combination of $\alpha_{i_1,\cdots,i_p}\dop x^{i_1} \wedge \cdots \wedge \dop x^{i_p} \otimes (s_1, \cdots s_p)$.
\end{dfn*}

\begin{dfn*}
  The exterior covariant derivative is the map given by extension of
  \begin{align*}
    \dop^{\nabla} \colon \Omega^p(M,E) & \to \Omega^{p+1}(M,E),\\
    \alpha \otimes u & \mapsto \dop^{\nabla}(\alpha \otimes u) = \dop \alpha \otimes u + (-1)^p \alpha \wedge \nabla u
  \end{align*}
  for $\alpha \in \bigwedge^p\T^{*}M$, $u \in \Omega^0(E)$.
\end{dfn*}

We want to define an inner product on $\Omega^p(M,E)$.
For this, fix a metric $g$ on $M$ and let $(E,\nabla,q) \to M$ be a vector bundle with metric and connection over $M$.
\begin{align*}
  \left< \alpha \otimes u, p \otimes v\right>
  = \int_M g(\alpha,p) q(u,v) \dop v_g
\end{align*}
is a number.
(For this integral to be finite, assume $M$ is compact or work with compactly supported sections.)

\begin{dfn*}
  The \CmMark{exterior covariant codifferential}\footnote{non-standard notation} is the formal $L^2$-adjoint of $d$
  \begin{align*}
    \delta^{\nabla} \colon \Omega^p(M,E) \to \Omega^p(M,E)
  \end{align*}
  such that $\left< \dop^{\nabla}(\alpha \otimes u), \beta \otimes v\right> = \left<\alpha \otimes u, \delta^{\nabla}(\beta \otimes v)\right>$.
\end{dfn*}

\begin{rem*}[Fact]
  An integration by parts arguement shows that $\delta^{\nabla}$ exists and, when $\nabla$ is a metric connection, then
  \begin{align*}
    \delta^{\nabla} \colon \Omega^1(M,E) \to \Omega^0(M,E),
    \
    \alpha \otimes u \mapsto -\tr_g(\nabla^{\T^{*} \otimes E} \alpha \otimes u),
  \end{align*}
  where for $\Omega^1(M,E) \to \Omega^0(M, \T^{*}M \otimes \T^{*}M \otimes E)$, we can take a trace with the metric by choosing an orthonormal basis.
\end{rem*}

\begin{dfn*}
  A \CmMark{harmonic $p$-form} with values in $E$ is an element $\omega_i \in \Omega^p(M,E)$ such that $\delta^{\nabla} = \delta^{\nabla} \omega = 0$.
  As a matter of fact this is equivalent to $\Delta \omega = 0$ for $\Delta := \delta^{\nabla} \circ \dop^{\nabla} + \dop^{\nabla} \circ \delta^{\nabla}$ (Consider $\left<\Delta \omega, \omega\right>$ and utilize the obvious stuff).
\end{dfn*}


\subsection{Definition of harmonic maps of 1st variation formula}

Let $(M,g)$ and $(N,h)$ be two riemannian manifolds and let $f \colon M \to N$ be a smooth map.
Then $\dop f \colon \T M \to \T N$ is an element $\dop f \in \Omega^0(\Hom(\T M, f^{*}\T N)) = \Omega^0(\T^{*}M \otimes f^{*}\T N)$.

Next, the metrics $g,h$ induce a metric on $\T^{*}M \otimes f^{*}\T N$.

\begin{dfn*}
  The energy density of $f \colon M \to N$ is $e(f) := \frac{1}{2} \left< \dop f, \dop f\right>_{\T^{*}M \otimes f^{*}\T N} = \frac{1}{2} \|\dop f\|^2$.
\end{dfn*}

Choose co-ordinates $\{x^i\}$ in $M$ and $\{y^i\}$ in $N$.
With respect to these, we have
\begin{align*}
  \frac{1}{2} \|\dop f \|^2 = \frac{1}{2}y^{ij} \partial_if^{*}\partial_jf^{\beta}h_{\alpha\beta}(f).
\end{align*}

\begin{dfn*}
  The \CmMark{Dirlichlet energy} is given by
  \begin{align*}
    E \colon C^2_0(M,N) \to \R,
    \
    f \mapsto \int_M e(f) \dop V_g.
  \end{align*}
  A \CmMark{critical map} (or \CmMark{stationary map}) is a map $f \colon M \to N$ such that for all compactly supported $F \colon M \times (-\varepsilon, \varepsilon) \to N$ $C^2$-map (variation of $f$) with $F(x,0) = f(x)$ we have that
  \begin{align}\label{eq:first-variation}
    \delta E(\nu) := \left.\frac{\dop}{\dop t} E(F) \right|_{t = 0} = 0
  \end{align}
  for $\nu = \frac{\dop}{\dop t} F|_{t = 0} \in \Omega^0(f^{*}\T N)$.
  The \cref{eq:first-variation} is called \CmMark{first variation in the direction of $\nu$}.
\end{dfn*}

\begin{dfn*}
  The map $f \colon (M,g) \to (N,h)$ is called \CmMark{harmonic}, if it is a critical point for the Dirlichlet energy.
\end{dfn*}

\begin{dfn*}
  Let $\dop f \in \Omega^1(M, f^{*}\T N)$ then $\nabla \dop f \in \Omega^0(M, \T^{*}M \otimes \T^{*}M \otimes E)$.
  
  The \CmMark{second fundamental form} of $f$ is $\nabla \dop f := B_f$, which is a symmetric $2$-tensor on $M$.
\end{dfn*}

\begin{dfn*}
  The \CmMark{tension field} of $f$ is the trace of $B_f$: $\tau(f) := \tr_g(B_f) \in \Omega^0(M,f^{*}\T N)$.
\end{dfn*}

\begin{thm*}[1st variation of $E$]
  Let $F \colon M \times (\varepsilon, \varepsilon) \to N$ a variation of $f$ and let $\nu = \frac{\dop}{\dop t}F|_{t = 0}$.
  Then
  \begin{align*}
    \delta E(\nu) = \frac{\dop}{\dop t}E(F)|_{t = 0} = - \int_M \left<\tau(f), \nu\right> \dop v_g.
  \end{align*}
\end{thm*}

\begin{proof}
  The variation $F \colon M \times (-\varepsilon,\varepsilon) \to N$ yields a pullback connection on $F^{*}\T N$, which shows
  \begin{align*}
    \frac{\dop}{\dop t}E(F)|_{t = 0} & = \frac{1}{2} \int_M \frac{\dop}{\dop t}\left<\dop F, \dop F\right> \dop V_g|_{t = 0}
    = \int_M \left<\nabla_{\frac{\partial}{\partial t}}\dop F, \dop F\right> \dop V_g|_{t = 0}\\
    & = \int_M \left<\nabla^{f^{*}\T N}\nu, \dop f\right> \dop V_g
    \overset{(*)}{=} \int_M \left<\nu, \delta^{\nabla^{f^{*}\T N}} \dop f\right> \dop V_g\\
    & = -\int_M \left< \nu, \tr_g(\nabla \dop f) \right> \dop V_g
    = - \int_M \left< \nu, \tau(f)\right> \dop V_g,
  \end{align*}
  where $(*)$ follows by a calculation in local co-ordinates.
\end{proof}

\begin{cor*}[Fundamental theorem of the calculus of variations]
  A $C^2$-map $f \colon (M,g) \to (N,h)$ is harmonic if and only if $\tau(f) = 0$.
\end{cor*}

What does $\tau(f) = 0$ look like?

Fix local co-ordinates $\{x^i\}$ on $M$ and $\{y^j\}$ on $N$.
Then $\dop f = \partial_if^{alpha} \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}}$ and thus
\begin{align*}
  \nabla \dop f
  & = \nabla \partial_i f^{\alpha} \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}}
  = \partial_j\partial_i f^{\alpha} \dop x^j \otimes \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}}
  + \partial_if^{\alpha} \nabla \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}}\\
  & = A + \partial_i f^{\alpha}(\nabla \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}} + \dop x^i \otimes \nabla \frac{\partial}{\partial y^{\alpha}})\\
  & = A + \partial_i f^{\alpha}( -\Gamma^i_{jk} \dop x^i \otimes \dop x^k \otimes \frac{\partial}{\partial y^{\alpha}} + \dop x^i \partial_j f^{\beta} \Gamma^{\gamma}_{\alpha\beta} \frac{\partial}{\partial y^{\gamma}})\\
  & = \partial_i \partial_jf^{\gamma} \Gamma_{ij}^k \partial_k f^{\gamma} + \Gamma_{\alpha\beta}^{\gamma}(f) \partial_jf^{\alpha}\partial_if^{\beta)} \dop x^i \otimes \dop x^j \otimes \frac{\partial}{\partial y^j}.
\end{align*}
Thus $\tau(f) = (\Delta_gf^{\gamma} + \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})$.


\section{Example and the Bochner formula (a glimpse of rigidity)}

Recall that above we considered $C^2$-maps $f \colon (M,g) \to (N,h)$ with tension field
\begin{align*}
  \tau(f) := \tr_g(\nabla \dop f) = 0 \in \Omega^0(M,f^*\T N).
\end{align*}
In local co-ordinates $\{x^i\}$ on $M$ and $\{y^{\alpha}\}$ on $N$ this means \footnote{Use roman indices for the $M$ and greek ones for $N$.} 
\begin{align*}
  \tau(f)^{\gamma} \frac{\partial}{\partial y^{\gamma}} = (\Delta_g f^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})\partial_{\gamma} = 0,
\end{align*}
where $\tilde \Gamma$ are the Christoffel symbols on $(N,h)$.

\begin{expl*}
  \begin{enumerate}[label=\Roman*.]
  \item Let $(M,g) = (\R, \dop t^2)$ and let $\eta \colon \R \to (N,h)$.
  From above we know that the Laplace-Beltrami operator here reads
  \begin{align*}
    \Delta_gf = g^{ij} (\partial_i\partial_jf - \Gamma_{ij}^k\partial_kf)
    = g^{ij}(\partial_i\partial_jf)
    = \partial_t^2f
  \end{align*}
  for
  \begin{align*}
    \Gamma_{ij}^k = \frac{g^{km}}{2}(\partial_ig_{im} + \partial_jg_{im} - \partial_m(g_{ij})
  \end{align*}
  ($=0$ in $\{g_{ij}\}$ is constant)\todo{repair} and $\{g_{ij}\} = g_{11} = f(\partial_t,\partial_t) = \dop t^2(\partial_t,\partial_t) = 1$.
  Hence
  \begin{align*}
    \tau(\eta)^{\gamma}\partial_{\gamma} = (\ddot \eta^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(\eta) \dot\eta^{\alpha}\dot\eta^{\beta})\partial_{\gamma} = 0,
  \end{align*}
  which is if and only if $\eta$ is a geodesic, i.e. the covariant derivate along $M$ of the curves speed vanishes: $\frac{\Dop}{\dop t} \dot \eta = 0$ and thus $E(\eta)|_a^b = \frac{1}{2}\int_a^b\|\dot\eta\|^2\dop t$.
\item Now let $f \colon (M,g) \to \R$.
  Here $\tau(f) = \Delta_gf = 0$.
  \begin{prop*}
    If $M$ is closed, then the energy harmonic functions are constant.
  \end{prop*}
  \begin{proof}
    By Green's theorem (integration by parts)
    \begin{align*}
      \int_M \underbrace{g(\nabla f, \nabla f)}_{=\|\nabla f\|^2} \dop V_g = - \int \Delta_g f \cdot f \dop V_g = 0
    \end{align*}
    for $\dop V_g = \sqrt{\det (\{g_{ij}\})} \dop x^1 \wedge \cdots \wedge \dop x^n$.
    Thus $\|\nabla f\|^2 = 0$ and $f$ must be constant.
  \end{proof}
  In our example, this shows $\Delta_g f = \lambda f$.
\item Let $f \colon (M,g) \to (N,h)$ be an isometric immersion, i.e. $\dop f$ is injective and $g = f^{*}h = h(\dop f, \dop f)$.
  Then we have
  \begin{align*}
    e(f) & = \frac{1}{2} \|\dop f\|^2
    = \frac{1}{2}h_{\alpha\beta} \partial_if^{\alpha}\partial_jf^{\beta}g^{ij}
    = \frac{1}{2}\partial_if^{\alpha}\partial_jf^{\beta}h(\partial_{\alpha},\partial_{\beta}) g^{ij}\\
    & = \frac{1}{2}h(\partial_if^{\alpha}\partial_{\alpha},\partial_jf^{\beta}\partial_{\beta})g^{ij}
    = \frac{1}{2}h(\dop f(\partial_i),\dop f(\partial_j)) g^{ij}
    = \frac{1}{2}g_{ij}g^{ij}
    = \frac{m}{2}
  \end{align*}
  and hence $E(f) = \frac{m}{2}\Vol(f)$, where $\Vol(f) = \int_M\dop V_{f^{*}h} = \int_M\dop V_g$.
  This shows that $f$ is critical for $E$ if and only if $f$ is critical for $\Vol \colon \Imm(M,N) \to \R_+$.
  The latter is clearly if and only if $f$ is a \textbf{minimal submanifold}.

  Examples of minimal submanifolds in $\R^3$ include the 2-plane, or the helicoid.
\end{enumerate}
\end{expl*}

\subsection{Composition laws for harmonic maps}

Consider the composition
\begin{align*}
  (M,g) \xrightarrow{f} (N,h) \xrightarrow{u} (Z,b).
\end{align*}
In general, if $f,u$ are harmonic, this needs not be harmonic again, which can be considered ``a bug or a feature''.
\begin{align*}
  B_{u \circ f}(X,Y) = B_u(\dop f(X), \dop f(Y)) + \dop u (B_f(X,Y))
\end{align*}
for $X,Y \in \T_pM$ and thus $B_{u \circ f} = \nabla^{\T^{*}M \otimes (u \circ f)^{*}\T N}(\dop(u \circ f))$.
Hence $\tau(u \circ f) = \dop (\tau(f)) + \tr_g(f^{*}B_u)$.

If $f$ is harmonic, then $\tau(u \circ f) = \tr_g(f^{*}B_u)$.
\begin{prop*}
  If $f \colon M \to N$, is harmonic and $u \colon N \to Z$ is totally geodesic, i.e. $B_u = 0$.
  Then $u \circ f$ is harmonic.
\end{prop*}

What if $u \colon N \to \R$ is a function and $f$ is harmonic?
Then
\begin{align*}
  \tau(u \circ f) = \tr_g(f^{*}B_u)
  = \tr_g(f^{*}(\Hess(u))
  = \sum_{i = 1}^n f^{*}(\Hess(u)) (E_i,E_i).
\end{align*}
Recall that a function $u \colon (N,h) \to \R$ is convex, if $\Hess(u)$ is positive definite.
If $f$ is harmonic and $u$ is convex, then $\tau(u \circ f) = \nabla_g u \circ f \geq 0$ (these are called \CmMark{subharmonic functions}).

\begin{thm*}
  A map is harmonic if and only if it pulls back germs of convex functions to germs of subharmonic functions.
\end{thm*}

There are various useful applications of the ``synthetic view'' on harmonic functions (e.g. Gromov-Shane).

\begin{thm*}
  Suppose $(M,g)$ is closed, connected and $(N,h)$ is $1$-connected with non-positive curvature.
  Then every harmonic map $f \colon (M,g) \to (N,h)$ is constant.
\end{thm*}

\begin{proof}
  The distance function $N \to \R_{\geq 0}, x \mapsto \dop_N(p,x)^2$ for every $p \in N$ is actually smooth and strictly convex, e.g. $\dop_{\R^n}(0,x)^2 = x_1^2 + \cdots + x_n^2$.

  In case $f$ is harmonic, we have
  \begin{align*}
    \Delta_gu \circ f = \tau(u \circ f) = \tr_g(f^{*}B_u) \geq 0
  \end{align*}
  and
  \begin{align*}
    -\int \| \dop(u \circ f)\|^2 \dop V_g = \int_M \Delta_gu \circ f \dop V_g \geq 0.
  \end{align*}
  Thus $\|\dop (u \circ f)\| = 0$ and hence $u \circ f$ is constant.
\end{proof}


\subsection{Bochner formulas}

Let $(E,\nabla,a)$ be a riemannina vector bundle, i.e. $a$ is a metric on $E$, $\nabla$ is a connection on $E$ preserving $a$ ($\nabla a = 0$) and there is a vector bundle projection map $E \to (M,g)$.

Let $\omega \in \Omega^p(M,E)$ and let $\nabla$ be a connection on $\Omega^p(M,E)$.
\begin{align*}
  \hat\nabla \colon \Omega^p(M,E) \to \Omega^p(M,\T^{*}M \otimes \T^{*}M \otimes E),
  \quad
  \omega \mapsto ( (X,Y) \mapsto \nabla_X\nabla_Y\omega - \nabla_{\nabla_XY}\omega )
\end{align*}
The \CmMark{trace Laplacian} is the operator
\begin{align*}
  \nabla^2 \colon \Omega^p(M,E) \to \Omega^p(M,E),
  \quad
  \omega \mapsto \tr_g(\hat\nabla \omega).
\end{align*}
Recall that the \CmMark{Hodge Laplacian} was the operator
\begin{align*}
  \dop^{\nabla} \colon \Omega^p(M,E) \to \Omega^{p+1}(M,E),
  \quad
  \alpha \otimes u \mapsto \dop \alpha \otimes u + (-1)^p \alpha \wedge \nabla u.
\end{align*}
With respect to the $L^2$-pairing $\beta \otimes v \mapsto \int_Mg(\alpha,\beta) a(u,v)\dop V_g$ it has a formal adjoint
\begin{align*}
  \delta^{\nabla}\colon \Omega^{p+1}(M,E) \to \Omega^p(M,E).
\end{align*}
The Hodge Laplacian is the degree preserving operator given by $\dop^{\nabla} \circ \delta^{\nabla} + \delta^{\nabla} \circ \dop^{\nabla} =: \Delta_a$.

The \CmMark[Bochner-Lichnerowicz formula]{(generalized) Bochner-Lichnerowicz formula} is given by
\begin{align*}
  \nabla_a \omega = - \nabla^2\omega + S_{\omega}.
\end{align*}
for $S_{\omega} \in \Omega^p(M,E)$ with
\begin{align*}
  S_{\omega}(X_1, \cdots, X_p) = \sum_{k = 1}^p\sum_{i = 1}^m(-1)^k(R^{\tilde \nabla}(e_i, X_k)\omega) (e_i,X_1, \ldots, \hat X_k, \ldots, X_n)
\end{align*}
for $X_i \in \T_pM$, $m = \dim M$ and $\{e_i\}$ an orthonormal frame around $p$.\footnote{Hat ($\hat X_k$), as always, means to omit the k-th term.}

\begin{cor*}
  Let $f \colon (M,g) \to (N,h)$ be harmonic.
  Then
  \begin{align*}
    \Delta_ge(f) = \|B_f\|^2 - \sum_{ij}\underbrace{h(R^h(f_{*}e_i,f_{*}e_j) f_{*}e_j, f_{*}e_i))}_{= \lambda \sec(e_i,e_j)} + \sum_ih(f_{*}(\Ric^g(e_i)),f_ke_i)
  \end{align*}
  for an orthonormal frame $\{e_i\}$.
\end{cor*}
The key observation for an application of this is that, if $\Ric^g$ is a positive operator, then the latter sum is positive.

\begin{thm*}[Eells-Sampson]
  Let $(M,g)$ be a closed with non-negative Ricci curvature and let $(N,h)$ have non-positive sectional curvature.
  \begin{enumerate}[label=(\roman*)]
  \item Then any harmonic map $f \colon (M,g) \to (N,h)$ is totally geodesic, i.e. $\nabla \dop f = B_f = 0$.
  \item If $\Ric^g$ is positive at any point, then $f$ is constant.
  \item If the sectional curvature of $(N,h)$ is strictly negative, then $f$ is constant or $f(M)$ is closed geodesic.
  \end{enumerate}
\end{thm*}

\begin{proof}
  The first statement easily follows from the corollary and $\int_M \left< \nabla u, \nabla v\right> \dop V_g = - \int_M \Delta u \cdot v \dop V_g$.
  The second is also not that hard and the last requires some work.
\end{proof}

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