Differentiable Manifolds and Matrix Lie Groups

  • Gregory L. Naber
Part of the Texts in Applied Mathematics book series (TAM, volume 25)


If X is a topological manifold and (U 1, φ 1) and (U 2, φ 2) are two charts on X with U 1U 2 ≠ ø, then the overlap functions φ 1 ° φ -1 : φ 1 (U 1U 2) → (φ 1(U 1U 2) and φ 2 ° φ -1: φ 1 (U 1U 2) are necessarily homeomorphisms between open sets in some Euclidean space. In the examples that we have encountered thus far (most notably, spheres and projective spaces) these maps actually satisfy the much stronger condition of being C , that is, their coordinate functions have continuous partial derivatives of all orders and types (see Exercise 1.1.8 and (1.2.4)).


Vector Field Real Vector Space Differentiable Manifold Coordinate Expression Smooth Vector Field 


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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Gregory L. Naber
    • 1
  1. 1.Department of Mathematics and StatisticsCalifornia State University, ChicoChicoUSA

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