# Riemannian geometry

## Vector bundles

### Basic Definitions

A pseudobundle of rank $$m$$ is a pair $$\mathbf{E}=(M,\{E_{p}\}_{p\in M})$$ where $$M$$ is a smooth manifold and $$E_{p}$$ are $$m$$-dimensional vector spaces. We will denote by $$E=\bigsqcup_{p\in M}E_{p}$$ the total space and $$\pi_{E}:E\to M$$, $$e\in E_{p}\mapsto p$$ the projection

Let $$\mathbf{E}$$ be a vector bundle over $$M$$ and let $$U\subset M$$ be an open subset. Then $$\mathbf{E}|_{U}:=(U,\{E_{p}\}_{p\in U})$$ is called the restriction of $$E$$ to $$U$$.