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\title The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 15
\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 25, 1998
\enddate
\endtopmatter
\document
\vskip.25in
\leftline{{\bf Connections on Vector Bundles I}}
\vskip.25in
There are three points of view of connections:
\roster
\item %(1)
A connection is a device for computing derivatives of sections of a
vector bundle.
\item %(2)
A connection is a device for decomposing tangent spaces to points in
$E$ into \itemitem
{$\bullet$} Vertical directions (along fibers of of $E\to B$).
\itemitem
{$\bullet$} Horizontal directions (``parallel'' to
tangent directions to $B$).
\item %(3)
A connection is a device for comparing fibers of $E$ at different
points $b_1$ and $b_2$ by ``parallel transport along a curve''.
\endroster
\vskip.25in
Before going on, we give a brief overiew of how these points of view
are related.
We start with (1), that is, the problem of differentitating sections.
\vskip.25in
Given $f\in C^{\infty}(B,\R)$, we get $df$, a global $1$-form on $B$.
That is,
$$d:C^{\infty}(B,\R)\to\Omega^1(B)=\Omega^0(B,T^*B)$$ such that
$d:f\mapsto df$. Recall that $df_b:T_bB\to\R$ is
given by $df_b(X_b)=X(f)(b)=\left.\dsize\frac d{dt}f(\gamma (t))\right|_{t=0}$
if $X_b\sim [\gamma (t)]$.
Now we can do the similar thing for $f:B\to\R^n$:
Given $X_b\sim [\gamma(t)]$,
$\left.\dsize\frac d{dt}f(\gamma (t))\right|_{t=0}
=(df_{1_b}(X_b),\cdots,df_{n_b}(X_b))$.
So $df=(df_1,\cdots,df_n)$ measures variation of $f$ along
$\gamma(t)$, that is, $df\in\Omega^0(B,T^*B\otimes\underline{\R^n})$.
Thus we get $$d:C^{\infty}(B,\R^n)\to\Omega^0(B,T^*B\otimes\underline{\R^n})
=\Omega^1(B,\underline{\R^n}).$$
What about for $s:B\to E$, that is, $s\in\Omega^0(B,E)$?
How do we make sense of $\left.\dsize\frac d{dt}s(\gamma (t))\right|_{t=0}$?
{\bf Problem:} $$
\left.
\aligned s(\gamma(t))&\in E_{\gamma(t)}\\
s(\gamma(0))&\in E_{\gamma(0)}\endaligned
\right\}\quad
\text{ cannot be identified.}
$$
Therefore we cannot evaluate $s(\gamma(t))-s(\gamma(0))$.
If we had a way of ``lining up''/ ``identifying'' all $E_{\gamma(t)}$
along $\gamma(t)$ (to $E_{\gamma(0)}$), then we could measure
variation of $s$ along $\gamma(t)$.
Thus one way to solve the problem is by specifying how to transport
$E_b$ along $\gamma(t)$, that is, by defining {\bf parallel transport}.
Without being too specific, to define parallel transports along a path
in $B$ we need to specify how paths in $B$ should be lifted to a paths
in $E$ (The lifted paths are the so-called {\bf horizontal
lifts}). But this allows us to define a lifting of tangent vectors on
$B$ to tangent vectors on $E$. That is, we get an identification of
$T_bB$ with a subspace $H_e$ of $T_eE$ for any $e\in E_b$. The
subspace $H_e$ is a complementary summand in $T_eE$ to the subspace of
vertical vectors, that is, we get a splitting $T_eE=V_e\oplus H_e$.
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