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\title       The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 8
\endtitle
\author      Professor Steven Bradlow \\ Class Notes From Math 433
\endauthor
\affil       University of Illinois at Urbana-Champaign
\endaffil
\address     273 Altgeld Hall, 1409 W. Green Street, Urbana, IL  61801
\endaddress
\email       bradlow\@math.uiuc.edu
\endemail
\date        February 9, 1998
\enddate

\endtopmatter

\document
Recall from last time that given
 $$E=(\coprod U_\alpha \times \Bbb R^n)/\{g_{\alpha \beta}\},$$
we can use $\{(g^t_{\alpha \beta})^{-1}\}$ to define $ E^* =(\coprod
U_\alpha \times \Bbb R^n)/\{(g^t_{\alpha \beta})^{-1}\}$.
Then $E^*_b =\text{Hom}(E_b,\Bbb R). $
All we need: If a vector space $V$ has a basis $\{v_1,\dots ,v_n \},$
then we can define the dual basis $\{v_1^*,\dots ,v_n^* \}$  for
 $V^*=\text{Hom}(V,\Bbb R)$, where $v_i^*:V \to \Bbb R$ defined by 
$$v_i^*(v_j)=\delta_{ij}.$$
Then any $\lambda \in V^*$ is given by $\sum_i \lambda_i v_i^*$.

For complex bundles:
 $$E=(\coprod U_\alpha \times \Bbb C^n)/\{g_{\alpha \beta}\}$$
and
$$g_{\alpha \beta}: U_\alpha \cap U_\beta \to \text{GL}(n,\Bbb C).$$
Exactly the same proof shows that $\{{g_{\alpha \beta}^t}^{-1}\}$ defines
$E^*$ and we can identify $\text{Hom}(\Bbb C^n ,\Bbb C) \cong \Bbb
C^n $ in the same way and see that $E_b^*=\text{Hom}(E_b ,\Bbb C)$.

\example{Remarks} 
There are some points to note:

\roster
\item %(1) 
We can use a Hermitian metric to identify 
$$\text{Hom}(\Bbb C^n ,\Bbb C)\overset\simeq\to\longrightarrow \Bbb
C^n,$$
but this is NOT a $\Bbb C$-linear map, so it is NOT the identification
we want!

\item %(2) 
Real case: Suppose that we can pick a local trivialization 
$E|_{U_\alpha} \overset\simeq\to\to U_\alpha \times \Bbb R^n $ such that
 $$g_{\alpha \beta}: U_\alpha \cap U_\beta \to \text{O}(n) \subset
\text{GL}(n,\Bbb R).$$
(so structure groups of $E$ can be reduced to $\text{O}(n)$)
Then $$g_{\alpha \beta}^t  g_{\alpha \beta} =I. $$
So $${g_{\alpha \beta}^t}^{-1}= g_{\alpha \beta}. $$
Conclusion: $$ E^* \cong E.$$

\item %(3) 
Complex case: Even if
 $$g_{\alpha \beta}: U_\alpha \cap U_\beta \to \text{U}(n) \subset
\text{GL}(n,\Bbb C).$$
Then $${\overline {g_{\alpha \beta}}}^t  g_{\alpha \beta}=I. $$
So $${g_{\alpha \beta}^t}^{-1}=\overline {g_{\alpha
\beta}}\ne{g_{\alpha \beta}}.$$
Therefore $E^* \ncong E$ as complex bundles. In fact $E^*\cong
\overline E$.
\endroster
\endexample

\vskip.25in
\leftline{{\bf Metrics on Bundles}}
\vskip.25in

Suppose that $g_{\alpha \beta}: U_\alpha \cap U_\beta \to \text{O}(n)$
for $E=(\coprod U_\alpha \times \Bbb R^n)/\{g_{\alpha \beta}\}$.

\proclaim{Claim}
We can define a fiberwise inner product on each $E_b$.  
\endproclaim
 
{\bf Proof:} Define $\langle \cdot\, ,\cdot \rangle_b$ on $E_b$, for each
 $b\in B$ as follows:
Let $( \cdot\, ,\cdot )$ be the standard inner product on
$\Bbb R^n$, that is, $( e_i,e_j)=\delta_{ij}$, where $\{ e_i\}$ is the 
standard basis for $\Bbb R^n$. Then we can identify $E_b\overset
\simeq^{\Psi_\alpha}\to\longrightarrow\Bbb R^n$ for any
$\alpha$ such that $b \in U_\alpha $ and define 
$e_i^\alpha (b)=\Psi_\alpha^{-1}(b, e_i)$.
\remark{Exercise 1} Show that $\{ e_i^\alpha (b)\}$ is a basis for $E_b$.
\endremark 
Now we define $\langle \cdot\, ,\cdot \rangle_b$ by declairing 
$\{ e_i^\alpha (b)\}$ to be an orthonormal frame, that is, by defining
$$\langle e_i^\alpha(b), e_j^\alpha(b)\rangle_b=\delta_{ij}.$$ 
Thus if $v=\Sigma v_i e^{\alpha}_i(b)$ and $u=\Sigma u_i
e^{\alpha}_i(b)$, then 

$$\split
\langle u, v\rangle_b 
   &=(\Psi_\alpha(b,v_1),\Psi_\alpha(b,v_2) )\\
   &=\sum_{i=1}^n u_i v_i.\endsplit$$ 
\qed
\remark{Exercise 2} Show that this is well-defined 
if $g_{\alpha \beta}\in\text{O}(n)$. 
\endremark 

\remark{Exercise 3} Show that if $\sigma_1$ and $\sigma_2$ 
are $C^{\infty}$ sections, 
then $b\mapsto\langle\sigma_1(b), \sigma_2(b)\rangle_b$ is a smooth map.
\endremark

\remark{Exercise 4} Suppose that $\Psi: E|_U\overset\simeq\to\to
U\times\Bbb R^n$ is a local trivialization and that we define 
$\{f_i(b)=\Psi^{-1}(b, e_i) \}_{i=1}^n$. (This is a basis for $E_b$ for
every $b\in U$ and $\{f_i\}$ is called a {\bf local frame}.) 

Define $$H_{ij}(b)=\langle f_j(b), f_i(b)\rangle_b,$$
show that the map $b\mapsto H_{ij}(b)$ defines a smooth map from $U$
to $\text{GL}(n)$.
\endremark 


\example{Remark} An assignment of an inner product 
$\langle \cdot\, ,\cdot \rangle_b$ to each fiber of $E$ is called a
{\bf smooth bundle metric} if either (hence both!) of the maps in
Exercises 3 and 4 are smooth.
\endexample

\remark{Exercise 5} Suppose $E$ admits a smooth bundle metric, that is, 
$\langle \cdot\, ,\cdot \rangle_b$ on each $E_b$, smoothly varying
with~$b$.
Show that we can pick $E|_U\overset\simeq\to\to
U\times\Bbb R^n$ such that $g_{\alpha \beta}: U_\alpha \cap U_\beta\to
\text{O}(n)$.
\endremark 

\example{Examples of existence of metrics}

\itemitem{\bf 1.} On $$\harrowlength=.42\sarrowlength
\commdiag{
B\times\Bbb R^n \cr
\mapdown \cr
\Bbb R^n}$$
Take $\langle\cdot\, ,\cdot\rangle_b=(\cdot\, ,\cdot)$ (or any other
metric on $\Bbb R^n$.)

\itemitem{\bf 2.} On $E=\coprod U_\alpha\times\Bbb R^n$,
take a partition of unity subordinate to $\{U_\alpha\}$---Need $\{U_\alpha\}$ 
locally finite here!
That is, $\{ \rho_\alpha: U_\alpha\to\Bbb R\ \text{is smooth}\}$ such that 
$\text{Supp}(\rho_\alpha )\subset U_\alpha$ and 
$\Sigma_\alpha \rho_\alpha(b)=1,\forall \ b$.
At $b\in U_\alpha$, we can define $\langle \cdot\, ,\cdot\rangle_{b,\alpha}$
using $\Psi_\alpha: E|_{U_\alpha}\overset\simeq\to\to
U_\alpha\times\Bbb R^n$.
Then we can set $\langle \cdot\, ,\cdot\rangle_b
=\Sigma_\alpha \rho_\alpha(b)\langle \cdot\,,\cdot\rangle_{b,\alpha}$. 
\endexample

\remark{Exercise 6} Show that this definition defines an inner product.
\endremark

\remark{Note}
Similarly for Hermitian metrics on complex vector bundles. Replacing
``orthonormal'' by ``unitary'' and $\text{O}(n)$ by $\text{U}(n)$, all
the expeted analogs of the above results apply.
\endremark

\remark{Exercise 7} Fill in the details.
\endremark


\enddocument
\end





