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\define\GL{\operatorname{GL}}
\define\ORTH{\operatorname{O}}
\define\Hom{\operatorname{Hom}}
\define\Iso{\operatorname{Iso}}
\define\End{\operatorname{End}}
\define\Aut{\operatorname{Aut}}
\define\Ker{\operatorname{Ker}}
\define\Coker{\operatorname{Coker}}
\redefine\Im{\operatorname{Im}}
\define\Rank{\operatorname{Rank}}


\define\R{\Bbb R}
\define\C{\Bbb C}



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\title       The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 16
\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 27, 1998
\enddate

\endtopmatter

\document






\vskip.25in
\leftline{{\bf Connections on Vector Bundles II}}
\vskip.25in

We now begin a detailed treatment of connections, starting with:
\definition{Definition 1}  For a vector bundle $E\to B$, 
a {\it connection} is a map 
$$ D:\Omega^0(B,E)\to\Omega^0(B,T^*B\otimes E) $$
such that (i) $D$ is linear and (ii) $D(fs)=df\otimes s+fDs$, 
for $f\in C^{\infty}(B,\R)$ and $s\in\Omega^0(B,E)$.
\enddefinition

\proclaim{Lemma } (ii) guarantees that $D$ is a local operator, that is, 
$Ds(b)$ depends only on $s$ ``near $b$''.
\endproclaim
{\bf Proof:}  Take a open neighborhood $U$ of $b$ and define a smooth bump
function $f$ on $U$ such that $f\equiv 1$ on $V(\subset U)$ near $b$
and $f\equiv 0$ outside $U$.
Then $D(fs)=df\otimes s+fDs$ implies $D(fs)(b)=Ds(b)$, but $fs$ is zero
outside $U$!
\qed

Hence local descriptions of $D$ are possible, that is, we can use local
frames to get a local description of $D$.
Say $E|_U\overset\simeq^{\Psi}\to\longrightarrow U\times\R$ is a local
trivialization, with corresponding local frame $\{ e_i\}^n_{i=1}$.  
Thus $e_i(b)=\Psi^{-1}(b,\hat {e_i})$, for $\hat {e_i}$ a standard basis
element for $\R^n$.
Then, over $U$, with respect to this section, $s$ has a local
description $s=\sum s_i e_i$ with $s_i:U\to\R$.
Applying (i) and (ii) in Definition 1, we can thus write 
$Ds=\sum ds_i\otimes e_i+s_i(De_i)$.

\proclaim{Claim }  $$De_i=\sum_j A_{ji}\otimes e_j $$ where $A_{ji}$
are $1$-forms defined on $U$. 
\endproclaim
{\bf Proof:}  $De_i$ is a (local) section of $T^*U\otimes E$ and every
section of $T^*U\otimes E$ has the form $\sum_j A_{ji}e_j$.
(Note. $A_{ji}$ depend only on $D$ and $\{ e_i\}$, NOT on $s$!)

Then $$Ds=\sum_{i,j} (ds_j+A_{ji}s_j)\otimes e_i.$$
If we identify $s=(s_1,\cdots ,s_n)^t$,
we see that 
$$D\left( \matrix 
s_1\\
\vdots\\
s_n
\endmatrix \right)
=(d+A)\left( \matrix 
s_1\\
\vdots\\
s_n
\endmatrix \right),$$
 where $A=[A_{ji}]$ is an $n\times n$ matrix of $1$-forms on $U$.
That is, $D=d+A$.\qed

\remark{Note} If $E=B\times\R^n$, then only one patch is required, that
is, we can pick $\{e_i\}$ to be a global frame. In this case, any
matrix of $1$-forms $A$ will produce a connection $D=d+A$.
In particular, $A\equiv 0$ yields $D=d$ (The difference between $D=d$
and $D=d+A$ will be clear later).
\endremark

\vskip.25in
If $E=\coprod U_{\alpha}\times\R^n/\{g_{\alpha \beta}\}$ is not
trivial and $\{e^{\alpha}_i\}$ is a local frame on $U_{\alpha}$,
then we must investigate compatibility of local descriptions:

Suppose $D=d+A^{\alpha}$ over $U_{\alpha}$ and $D=d+A^{\beta}$ over
$U_{\beta}$, then $$De^{\alpha}_i=A^{\alpha}_{ji}e^{\alpha}_j.\tag 1$$
Also $e^{\alpha}_i=g^{\alpha\beta}_{ji}e^{\beta}_j $, so 
$$\split
   De^{\alpha}_i
   &=D(g^{\alpha\beta}_{ji}e^{\beta}_j)\\
   &=dg^{\alpha\beta}_{ji}\otimes
     e^{\beta}_j+g^{\alpha\beta}_{ji}De^{\beta}_j\\
   &=dg^{\alpha\beta}_{ji}g^{\beta\alpha}_{kj}\otimes e^{\alpha}_k
     +g^{\alpha\beta}_{ji}A^{\beta}_{kj}g^{\beta\alpha}_{lk}
      \otimes e^{\alpha}_l\\
   &=\Bigl( \left[g^{\beta\alpha}\cdot dg^{\alpha\beta}\right]_{li}
            +\left[g^{\beta\alpha} A^{\beta} g^{\alpha\beta}\right]_{li}\Bigr)
     \otimes e^{\alpha}_l.\endsplit\tag 2
$$
Therefore the compatibility implies (1)=(2). Hence 

$$ A^{\alpha}=g^{\beta\alpha} A^{\beta} g^{\alpha\beta}
               +g^{\beta\alpha}\cdot dg^{\alpha\beta}.\tag $\ast$ $$
\vskip.25in
\noindent
\underbar{Conclusion}: $D$ is specified by $\{A^{\alpha}\}$, where 
$A^{\alpha}$ is a matrix value $1$-form on $U_\alpha$ and $A^{\alpha}$
and $A^{\beta}$ related by ($\ast$) on $U_\alpha\cap U_\beta$.

\remark{Note} We cannot take $A^{\alpha}\equiv 0$, because
this does not in general satisfy ($\ast$).

\noindent
\underbar{Exception}: If $dg^{\beta\alpha}=0$, that is, 
$g^{\beta\alpha}: U_\alpha\cap U_\beta\to\GL(n)$ is (local) constant. 
For reason which will become clear, such bundles are called {\bf flat
bundles}:
\endremark

\definition{Definition 2} If $E$ admits description 
$E=\coprod U_{\alpha}\times\R^n/\{g_{\alpha \beta}\}$ such that 
$dg^{\alpha\beta}=0$, then $E$ is called {\it flat}.
\enddefinition
If $E$ is a flat bundle, we can define a connection $D=d$ with 
respect to corresponding flat local frames.




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