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

Vector Field Open Subset Tangent Space Tangent Vector Open Neighborhood
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Copyright information

© Springer Science+Business Media New York 2013