09-10 Notes

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\title{MIT Geometry Seminar \\ Lecture 3 \\ 9/10/2008}

\maketitle

Let $X$ be a compact Riemanian manifold, and $P$ be a principal bundle over $X$ with structure group $G$, associated to a vector bundle $E \rightarrow X$. We denote the space of $L_k^p$ connections by $\mathcal{A}_k^p$, more precisely, $\mathcal{A}_k^p$ = $\{$ A connection $|$ if $\tau: \pi^{-1}(U) \rightarrow U \times P$ is a $C^\infty$ trivialization, $\varphi \in C_o^\infty(U)$, and $a_\tau \in \Omega^1(U,\mathfrak{g})$, then $\varphi\alpha_\tau \in L_k^p(U, T^*U \otimes \mathfrak{g}) \}$. We denote the $L_{k+1}^p$ gauge transformations by $\mathcal{G}_{k+1}^p$, more precisely $\mathcal{G}_{k+1}^p$ = $\{ u \in L_{k+1}^p$($X$, End $E$) $| u^*u = 1$ a.e. $\}$. Last time, we noted that we have a smooth action $\mathcal{G}_{k+1}^p \times \mathcal{A}_k^p \rightarrow \mathcal{A}_k^p$ provided $p(k+1) > n$, we denote the quotient of this action by $\mathcal{B}_k^p$ = $\mathcal{A}_k^p / \mathcal{G}_{k+1}^p$.

In light of the Sobolev Embedding theorems, we can ask the following question: does an embedding $\mathcal{A}_k^p \hookrightarrow \mathcal{A}_l^q$ yield a map $\mathcal{B}_k^p \hookrightarrow \mathcal{B}_l^q$?

\begin{prop}

Suppose $u \in \mathcal{G}_0^\infty = \{ u \in L^\infty(X, End E) | u^*u = 1$  a.e. $\}$ and $\exists A, \tilde{A} \in \mathcal{A}_k^p$ such that $u\cdot A = \tilde{A}$, then $u \in \mathcal{G}_{k+1}^p$.

\end{prop}

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