Lie Groups pp 39-43 | Cite as

# Vector Fields

Chapter

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## Abstract

A *smooth premanifold* of dimension *n* is a Hausdorff topological space *M* together with a set \(\mathcal{U}\) of pairs (*U*,ϕ), where the set of *U* such that \((U,\phi ) \in \mathcal{U}\) for some ϕ is an open cover of *M* and such that, for each \((U,\phi ) \in \mathcal{U}\), the image ϕ(*U*) of ϕ is an open subset of \({\mathbb{R}}^{n}\) and ϕ is a homeomorphism of *U* onto ϕ(*U*). We assume that if \(U,V \in \mathcal{U}\), then \(\phi _{V } \circ \phi _{U}^{-1}\) is a diffeomorphism from \(\phi _{U}(U \cap V )\) onto \(\phi _{V }(U \cap V )\). The set \(\mathcal{U}\) is called a *preatlas*.

### Keywords

Manifold## Copyright information

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