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 \(U\).