Frame Bundles and Spacetimes

  • Gregory L. Naber
Part of the Applied Mathematical Sciences book series (AMS, volume 141)


An n-dimensional smooth manifold X is locally diffeomorphic to ℝ n and the study of such manifolds generally necessitates piecing together well-understood local information about ℝ n into global information about X. The basic tool for this piecing together operation is a partition of unity. In this section we will prove that these exist in abundance on any manifold and use them to establish the global existence of two useful objects that clearly always exist locally (Riemannian metrics and connection forms). The same technique will be used in Section 4.3 to obtain a convenient reformulation of the notion of orientability while, in Section 4.6, a theory of integration is constructed on any orientable manifold by piecing together Lebesgue integrals on coordinate neighborhoods.


Open Cover Smooth Manifold Orthonormal Frame Riemannian Metrics Connection Form 
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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Gregory L. Naber
    • 1
  1. 1.Department of Mathematics and StatisticsCalifornia State UniversityChicoUSA

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