Lie Groups pp 31-37 | Cite as

Lie Subgroups of \(\mathrm{GL}(n, \mathbb{C})\)

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

Abstract

If U is an open subset of \({\mathbb{R}}^{n}\), we say that a map \(\phi: U\longrightarrow {\mathbb{R}}^{m}\) is smooth if it has continuous partial derivatives of all orders. More generally, if \(X \subset {\mathbb{R}}^{n}\) is not necessarily open, we say that a map \(\phi: X\longrightarrow {\mathbb{R}}^{n}\) is smooth if for each xX there exists an open set U of \({\mathbb{R}}^{n}\) containing \(x\) such that ϕ can be extended to a smooth map on U. A diffeomorphism of \(X \subseteq {\mathbb{R}}^{n}\) with \(Y \subseteq {\mathbb{R}}^{m}\) is a homeomorphism \(F: X\longrightarrow Y\) such that both F and F −1 are smooth. We will assume as known the following useful criterion.

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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