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Relativity in Modern Physics$
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Nathalie Deruelle and Jean-Philippe Uzan

Print publication date: 2018

Print ISBN-13: 9780198786399

Published to Oxford Scholarship Online: October 2018

DOI: 10.1093/oso/9780198786399.001.0001

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The covariant derivative and the curvature

The covariant derivative and the curvature

Chapter:
(p.647) 23 The covariant derivative and the curvature
Source:
Relativity in Modern Physics
Author(s):

Nathalie Deruelle

Jean-Philippe Uzan

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198786399.003.0063

This chapter first considers the tangent spaces of a non-connected manifold, in which the tangent t at the set of points p in the manifold is an element of the tangent space at p. Afterward, the chapter summarizes the elementary introduction to the exterior calculus of Chapter 5 of Book 2. Next, the chapter studies the Lie bracket and Lie derivative, before moving on to the covariant derivative and a connected manifold. The covariant derivative in particular is introduced to ensure the effectiveness of the Lie brackets and the Lie derivative. From here, this chapter considers the torsion of a covariant derivative and finally to the curvature of a covariant derivative.

Keywords:   covariant derivative, curvature of a covariant derivative, tangent spaces, exterior calculus, Lie bracket, Lie derivative, connected manifold, non-connected manifold

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