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  • Gerardo F. Torres del Castillo

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., vT x M and wT 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 
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.

References

  1. Hochstadt, H. (1964). Differential Equations: A Modern Approach (Holt, Rinehart and Winston, New York) (Dover, New York, reprinted 1975). Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Instituto de CienciasUniversidad Autónoma de PueblaPueblaMexico

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