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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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Riemannian manifolds

Riemannian manifolds

Chapter:
(p.656) 24 Riemannian manifolds
Source:
Relativity in Modern Physics
Author(s):

Nathalie Deruelle

Jean-Philippe Uzan

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198786399.003.0064

This chapter is about Riemannian manifolds. It first discusses the metric manifold and the Levi-Civita connection, determining if the metric is Riemannian or Lorentzian. Next, the chapter turns to the properties of the curvature tensor. It states without proof the intrinsic versions of the properties of the Riemann–Christoffel tensor of a covariant derivative already given in Chapter 2. This chapter then performs the same derivation as in Chapter 4 by obtaining the Einstein equations of general relativity by varying the Hilbert action. However, this will be done in the intrinsic manner, using the tools developed in the present and the preceding chapters.

Keywords:   Riemannian manifolds, metric manifold, Levi-Civita connection, curvature tensor, Riemann–Christoffel tensor, covariant derivative, Einstein equations, general relativity, Hilbert action

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