Lie Groups pp 39-43 | Cite as

Vector Fields

  • Daniel Bump
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)

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

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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