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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

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

Authors and Affiliations

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

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