Editing Lecture 6 - Thursday Week 3 (section)

== Vector Bundles ==

[[Vector Bundle]]s $\pi: E \rightarrow M$ and their sections $\Delta: M \rightarrow E$. We have the notation $\Gamma(E) = \{\Delta | \Delta \text{section of }E\}$

If $\dim(M) > 0$ then $\Gamma(E)$ is infinite-dimensional $\mathbb{R}$ vector space. Also $\Gamma(E)$ is a $C^\infty(M)$-module.

Prop (Ex): If $M$ paracompact, then for all vector bundles $\pi:E \rightarrow M$, there exists $\tilde{E} \rightarrow M$ (finite rank vector bundle) sch that $E \oplus \tilde{E} \rightarrow M$ is trivial. $E \oplus \tilde{E} \cong M \times k^N$ for some $N \in \mathbb{N}$.

def: an $R$-module $\epsilon$ is projective if there exists $\tilde{\epsilon}$ such that $\epsilon \oplus \tilde{\epsilon } \cong R^N$.

Theorem (Serre-Swan): vector bundles over $M$ $\leftrightarrow$ projective modules $M$ over $R = C^\infty(M)$.
\textit{Note: we won't really use this theorem.}

prop: If $E$, $F$ are vector vundles, then can take:

* direct sum $E \oplus F$ 
* duals $E^*$ 
* tensor product $E \otimes F$
* exterior product $\wedge^k E$
* symmetric products sym$^k E$

def: A subbundle $E \subset F$ is a vertical morphism $E \rightarrow F \rightarrow M$ that is injective everywhere. (note there shold be a commutative diagram here including the map $E \rightarrow M$)

e.g. if $f: M \rightarrow N$ embedding (or immersion) then $TM \hookrightarrow f^* TN$ is a subbundle over $M$.

prop: If $E \hookrightarrow F$ is a subbundle, then can take quotient bundle $F/E$.

e.g. if $f: M \rightarrow N$ immersion, then $\text{Norm}_{M/N} = f^*(TN)/TM$