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Gerardo F. Torres del Castillo ()
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Gerardo F. Torres del Castillo: Universidad Autónoma de Puebla, Instituto de Ciencias

Chapter Chapter 5 in Differentiable Manifolds, 2012, pp 93-114 from Springer

Abstract: Abstract The tangent space, T x M, to a differentiable manifold M at a point x is a vector space different from the tangent space to M at any other point y, T y M. In general, there is no natural way of relating T x M with T y M if $x \not= y$ . This means that if v and w are two tangent vectors to M at two different points, e.g., v∈T x M and w∈T y M, there is no natural way to compare or to combine them. However, in many cases it will be possible to define the parallel transport of a tangent vector from one point to another point of the manifold along a curve. Once this concept has been defined, it will be possible to determine the directional derivatives of any vector field on M; conversely, if we know the directional derivatives of an arbitrary vector field, the parallel transport of a vector along any curve in M is determined.

Keywords: Vector Field; Tangent Vector; Curvature Tensor; Directional Derivative; Tensor Field (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-0-8176-8271-2_5

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DOI: 10.1007/978-0-8176-8271-2_5

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