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\title The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 17
\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 March 2, 1998
\enddate
\endtopmatter
\document
\leftline{{\bf Existence of Connections}}
\vskip.25in
Say that a bundle is described by
$$E=(\coprod U_\alpha \times \Bbb R^n)/\{g_{\alpha \beta}\}$$
Here are two ways to construct a connection on $E$. Both methods use connections, $D_\alpha$ defined on $E|U_\alpha$, the local trivializations $\psi_\alpha:E|U_\alpha \to U_\alpha \times \Bbb R^n$ and a partition of
unity, $\{\rho_\alpha\}$ subordinate to the cover $\{U_\alpha\}$. The two connections are defined by
\itemitem{(a)} $D_1=\sum_\alpha D_\alpha \circ \rho_\alpha$
\itemitem{(b)} $D_2=\sum_\alpha \rho_\alpha D_\alpha$
\vskip.25in
\remark{Note} $D_\alpha$ was defined on $E|U_\alpha$ by:
\itemitem{(a)} Picking a connection, $\tilde D_\alpha$ on $U_\alpha \times \Bbb R^n$.
\itemitem{(b)} Setting $D_\alpha=\psi_\alpha^{-1} \circ \tilde D_\alpha \circ \psi_\alpha$, i.e, $D_\alpha(s)=\psi_\alpha^{-1}(\tilde D_\alpha(\psi_\alpha(s)))$
\endremark
\vskip.25in
\remark{Exercise 1} Show that this is a connection.
\endremark
\vskip.25in
In fact, given a bundle isomorphism, $h:E \to F$ and a connection $D$ on $F$, we define a connection, $h^*D$, on $E$ by $h^*D=h^{-1} \circ D \circ h$.
\vskip.25in
In the definition of $D_2$, the terms in the sum are of the form
$$(\rho_\alpha D_\alpha(s))(b)=\rho_\alpha(b) D_\alpha(s)(b)=\cases \rho_\alpha(b) D_|{\alpha(s)(b)} &b \in U_\alpha \cr 0 &\text{ otherwise} \endcases $$
\vskip.25in
\remark{Exercise 2} Check that $D_2$ does define a connection.
\endremark
\vskip.25in
We now examine the connection, $D_1$. We define
$$\aligned
(D_\alpha \circ \rho_\alpha)(s)&=D_\alpha(\rho \cdot \alpha(s)) \cr &=d \rho_\alpha \otimes s+\rho_\alpha D_\alpha(s) \endaligned $$
Using this, we see that $$D_1(s)=D_2(s)+\sum_\alpha d\rho_\alpha \otimes s$$
[Note that off of $U_\alpha$, $d \rho_\alpha=0$ since the support of $\rho_\alpha$ is in $U_\alpha$.]
\vskip.25in
\remark{Exercise 3} Show that $D_1$ defines a connection.
\endremark
\vskip.25in
The diffrerence between $D_1$ and $D_2$ is instructive.
\vskip.25in
\remark{Note} $D_1-D_2$ is a $\text{End}(E)$ valued 1-form. In fact
$$\aligned (D_1-D_2)&=\sum_\alpha d \rho_\alpha \otimes \text{Id}\cr
\text{i.e, for any }s \in \Omega^0(E)\quad (\sum_\alpha d\rho_\alpha \otimes \text{Id})(s)&=\sum_\alpha d\rho_\alpha \otimes s \endaligned $$
\endremark
\vskip.25in
This illustrates a general fact.
\proclaim{Proposition} If $D_1$ and $D_2$ are connections on a vector bundle,
$\pi:E \to B$, then $$\aligned
D_1-D_2 &\in \Omega^0(B, T^*B \otimes \text{End}(E)) \cr
&=\Omega^1(B,\text{End}(E)) \endaligned $$
\endproclaim
\vskip.25in
{\bf Proof:}
We check:
\itemitem{(a)} This is linear with respect to constants.
\itemitem{(b)} This is also linear with repsect to functions: $$\aligned (D_1-D_2)(fs)&=(df \otimes s+fD_1(s))-(df \otimes s+f D_2(s)) \cr
&=f(D_1-D_2)(s) \endaligned $$
\vskip.25in
\remark{Note} With respect to local frames of $E$, say $\{e_i^\alpha\}_i^{\text{rank E}}$, locally a section of $T^*M \otimes \text{End}(E)$ is a matrix of 1-forms, $A^\alpha$. Over $U_\alpha$, $A^\alpha$ and $A^\beta$ must be related by the transition functions,
$$A^\alpha = g_{\alpha \beta} A^\beta g_{\beta \alpha}$$ If $(D_1-D_2)(fs)=f(D_1-D_2)(s)$, then the local descriptions (with respect to local frames) will also have this property.
\endremark
\vskip.25in
\remark{Exercise 4} Repeat the computation in the proof of (b) for connection 1-forms.
\endremark
\vskip.25in
{\bf Summary:} $D_1-D_2 \in \Omega^1(B,\text{End}(E))$. Conversely, if $A \in \Omega^1(B,\text{End}(E))$, then we can define $(D_1+A)(s)=D_1(s)+A(s)$.
\vskip.25in
\remark{Exercise 5} Show that $D_1+A$ is a connection.
\endremark
\vskip.25in
{\bf Conclusion:} If $\Cal{A}(E)$ is the space of all connections on $E$, then $\Cal{A}(E)=D_0+\Omega^1(B,\text{End}(E))$ where $D_0$ is any fixed connection. That is, $\Cal{A}(E)$ is an infinite dimensional affinve space based on
$\Omega^1(E,\text{End}(E))$.
\vskip.25in
\leftline{{\bf Parallel Transport}}
\vskip.25in
\definition{Definition 1} Given a connection, $D$, and a vector field,
$X$, then $D_Xs$ is called the {\it covariant derivative} of $s$ along $X$.
We can write it locally: $D_Xs=d_Xs+(A(X))s$, where $D_xs \in \Omega^0(B,E)$ and $(D_Xs)(b)=(d_{X_b}s)+A(X_b)s(b)$, that is, it depends only on $X_b \in T_bB$.
\enddefinition
\definition{Definition 2} If $s \in \Omega^0(B,E)$ satisfies $Ds=0$, then
we say that $s$ is {\it parallel}.
\enddefinition
\remark{Question} Can we find solutions to $Ds=0$?
\endremark
With respect to a local frame, if
$$\displaystyle{
s = \pmatrix s_1\\
s_2 \\
\dots \\
s_n
\endpmatrix}$$
and $D=d+A_1$, then the condition for being parallel is $d s_i+A_{ij}s_j=0$ for all $i$. With local coordinates, $(x_1,\dots,x_k)$, on $B$, we are thus looking at trying to solve the system of equations, $$\sum_{\alpha=1}^k \frac{\partial s_i}{\partial x_\alpha}+(A_{ij}^\alpha s_j)dx_\alpha=0 \quad i=1,\dots,n$$ where $A_{ij}=A^\alpha_{ij}dx_\alpha$. This is a system of partial differential equations for which existence of solutions is NOT guarenteed.
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