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Covariant Derivatives and Geodesics

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Part of the Graduate Texts in Mathematics book series (GTM,volume 160)

Abstract

Throughout this chapter, by a manifold, we shall mean a C manifold, for simplicity of language. Vector fields, forms and other objects will also be assumed to be C unless otherwise specified. We let X be a manifold. We denote the R-vector space of vector fields by ΓT(X). Observe that ΓT(X) is also a module over the ring of functions \( \mathfrak{F} = \mathfrak{F}^\infty (X). \) We let π:TX →X be the natural map of the tangent bundle onto X.

Keywords

  • Vector Field
  • Riemannian Manifold
  • Covariant Derivative
  • Tangent Bundle
  • Local Representation

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.

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© 1995 Springer-Verlag New York, Inc.

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Lang, S. (1995). Covariant Derivatives and Geodesics. In: Lang, S. (eds) Differential and Riemannian Manifolds. Graduate Texts in Mathematics, vol 160. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4182-9_8

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  • DOI: https://doi.org/10.1007/978-1-4612-4182-9_8

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-8688-2

  • Online ISBN: 978-1-4612-4182-9

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