authorea.com/79725

Riemannian geometry

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 \(U\)*.

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