\classheader{2013-03-01}

Last time, we were discussing the Ehresmann theorem for fiber bundles of pairs. There was just one thing left to prove.

In the notation of the last lecture, we had $\Cinfty$ manifolds $X$ and $Y$, a closed $\Cinfty$ submanifold $A\subseteq X$, and a $\Cinfty$ map $f:X\to Y$ such that both $f$ and $f|_A$ are submersions. (Note that for the Ehresmann theorem, we would assume properness, but for now we just want to extract the subbundle $W$ which did not need that hypothesis.)

\begin{proposition}
There exists a subbundle $W\subset TX$ such that
\begin{itemize}
\item[(i)] For all $x\in X$, the derivative $f'(x)|_{W(x)}:W(x)\to T_{f(x)}Y$ is an isomorphism.
\item[(ii)] For all $x\in A$, we have $W(x)\subset TA$ (both interpreted as subspaces of $T_xX$).
\end{itemize}
\end{proposition}
This proposition implies the Ehresmann theorem for pairs.

The secret code phrase here is that
\begin{center}
$H^1(\text{any sheaf of modules over the sheaf of $\Cinfty$ functions})=0$
\end{center}
\begin{proof}
For the first step, note that the problem makes sense on any open $U\subset X$, so it will suffice to show that $W$ exists locally, i.e. that for all $x\in X$, there is a neighborhood $U(x)$ where the theorem holds.

If $x\notin A$, then we're done, so suppose that $x\in A$. WLOG, we can take $X=\R^n$, $A=\{x\in\R^n\mid x_{m+1}=\cdots=x_n=0\}$, and $f:X\to Y$ the map $f(x_1,\ldots,x_n)=(x_1,\ldots,x_r)$ where $r\leq m$. In this case, we can just take $W$ to be the span of $\frac{\partial}{\partial x_1},\ldots,\frac{\partial}{\partial x_r}$.

Now to Step 2; we want to provide an algebraic description of $W$. This is essential. We can't add subbundles, but we can add / do other linear things to sections of bundles.

For each $x\in X$, let $s(x)$ be the inverse of the isomorphism described in (i); in other words, we want to demonstrate the existence of a map of bundles $s:f^*TY\to TX$ such that
\begin{itemize}
\item[(i')] $f'(x)\circ s(x):T_{f(x)}Y\to T_{f(x)}Y$ is the identity for all $x\in X$
\item[(ii')] For all $x\in A$, we have $s(x)(T_{f(x)}Y)\subseteq T_xA$.
\end{itemize}

Step 3: Suppose that $s_1$ and $s_2$, both maps $f^*TY\to TX$, satisfy conditions (i') and (ii'). Then $h=s_2-s_1:f^*TY\to TX$ satisfies
\begin{itemize}
\item[(i'')] $f'(x)\circ h(x)=0$ for all $x\in X$
\item[(ii'')] $h(x)(T_{f(x)}Y)\subseteq T_xA$ for all $x\in A$
\end{itemize}
so that
\[Z=\{h:f^*TY\to TX\mid \text{(i'') and (ii'') hold}\}\]
is a module over the ring of $\Cinfty$ functions on $X$. Note that this is a characterization; in other words, if $s_1$ satisfies (i') and (ii'), then $s_1+h$ satisfies them if and only if $h\in Z$.

As a corollary of Step 3, we see that if $s_1,\ldots,s_m$ are as in Step 2, and $\varphi_1,\ldots,\varphi_m:W\to\R$ are a $\Cinfty$ partition of unity (so that $\sum \varphi_i=1$), then $\sum \varphi_is_i$ also satisfies the conditions of step 2, because
\[\sum \varphi_is_i=\underbrace{\sum \varphi_i(s_i-s_1)}_{\in\,Z}+\underbrace{\left(\sum \varphi_i\right)}_{=\,1}s_1.\]

Now we come to the proof of the proposition itself. Let $\{U_\lambda\}_{\lambda\in\Lambda}$ be an open cover equipped with $s_\lambda:f^*TY|_{U_\lambda}\to TX|_{U_\lambda}$ all satisfying (i') and (ii'). There is a partition of unity $\varphi_\lambda$ subordinate to $U_\lambda$; then $\varphi_\lambda s_\lambda$ (originally defined only on $U_\lambda$) can be extended by 0 to a $\Cinfty$ map $\varphi_\lambda s_\lambda:f^*TY\to TX$. Now let $s=\sum \varphi_\lambda s_\lambda: f^*TY\to TX$; the corollary above implies that $s$ satisfies (i') and (ii').
\end{proof}


\subsection*{Existence of inner products on vector bundles}

Given a $\Cinfty$ vector bundle $f:V\to M$, we want to construct a map $B:V\times_M V\to \R$ such that $B:V(x)\times V(x)\to \R$ is a positive definite, symmetric, bilinear form.

If $W$ is a vector space, and $B:W\times W\to\R$ is symmetric and bilinear, we say that $B$ is positive semi-definite if $B(w,w)\geq 0$ for all $w\in W$, and positive definite if it is positive semi-definite and $B(w,w)=0$ implies $w=0$.

\begin{proof}
Step 1. Assume that $V|_U$ is a trivial bundle, i.e. there exist sections $s_1,\ldots,s_k$ of $V|_U$ such that $s_1(x),\ldots,s_k(x)$ form a basis for $V(x)$ for all $x\in U$.

Define $B_U(s_i(x),s_j(x))=\delta_{ij}(x)$. Given an open cover $\cal{U}$, and a partition of unity $\varphi_U$ subordinate to $\cal{U}$, then $\sum \varphi_UB_U$ is a symmetric bilinear positive semi-definite form. But for any $x\in X$, if $v\in V(x)$ is non-zero, then there is some $U$ such that $\varphi_U(x)>0$, so that $x\in U$ and moreover $B_U(v,v)>0$, hence $B(v,v)\geq \varphi_U(x)B_U(v,v)>0$. Thus, this is in fact positive definite.
\end{proof}

\subsection*{Existence of connections on a vector bundle}

A good reference for this is Milnor's \textit{Morse Theory}.

Let $p:V\to M$ be a $\Cinfty$ vector bundle. A connection is essentially a way of taking a derivative of a section $s$ of a vector bundle $v$ with respect to a vector field on $M$.

Suppose that $x\in U$ and that $V|_U$ is trivial, and that $s_1,\ldots,s_k$ are sections of $V|_U$ that give a basis for $V(x)$ for each $x\in U$. For any $v\in T_xM$, we define
\[v\left(\sum f_is_i\right)=\sum v(f_i)s_i.\]
A connection, or a covariant derivative, $\connection$ on $V$ is a map taking in a vector field $v$ on $M$, and a section $s$ of $V$, and outputting $\connection_vs$, another section of $V$. We also require that a connection satisfy certain properties: for any $\Cinfty$ map $f:M\to\R$,
\begin{enumerate}
\item $\connection_v(s_1+s_2)=\connection_v(s_1)+\connection_v(s_2)$
\item $\connection_v(fs)=v(f)s+f\connection_v(s)$ (this is the Leibniz rule)
\item $\connection_{fv}(s)=f\connection_v(s)$
\end{enumerate}
We could have stated this definition sheaf-theoretically, which is after all necessary to do it on analytic manifolds, but for $\Cinfty$ manifolds, they are equivalent.

We want to show that any $\Cinfty$ vector bundle $V\to M$ has a connection.

The argument is the same as we've been doing. Step 1 is to show that they exist locally (this is just the trivial connection). Step 2 is to take two connections $\connection^1,\connection^2$ and define $h$ via $\connection^2=\connection^1+h$, i.e. $\connection^2_v(s)=\connection^1_v(s)+h_vs$ for all sections $s$, and note that $h$ satisfies three properties: $h$ is additive in $s$,
\[h_v(fs)=f h_v(s)\]
for all $\Cinfty$ functions $f$, and $h_{fv}(s)=f h_v(s)$.

Then, if $\connection^1$ is a connection and $\connection^2=\connection^1+h$, then $\connection^2$ is a connection if and only if $h$ satisfies the above three properties. The collection of all such $h$ can be thought of being comprised of precisely the sections of $\Hom(TM,\End(V))$, which is a module over $\Cinfty$ functions $M\to\R$.

We then conclude by using a partition of unity and noting that $\sum \varphi_U\connection_U$ gives a connection.

Let's examine connections in a basic case; let $M$ be an open interval $(a,b)$. By the properties of a connection, all we have to look at is $\connection_{\frac{d}{dt}}(s)$. In particular, what is
\[\{\text{sections }s:M\to V\mid \connection_{\frac{d}{dt}}(s)=0\}\quad ?\]
We know that $V$ is trivial because we're working on an interval; choose a specific trivialization, so that we will think of sections as maps $s:(a,b)\to\R^k$. Define vectors of $\Cinfty$ functions $m_i$ by
\[\connection_{\frac{d}{dt}}(e_i)=m_i,\]
where
\[e_i=\begin{bmatrix}
0 \\ \vdots \\ 1 \\ \vdots \\ 0
\end{bmatrix},\]
the 1 being in the $i$th position. Then