Difference between revisions of "Riemannian manifold"

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# A [[differential manifold]] <math>M</math>
 
# A [[differential manifold]] <math>M</math>
 
# A [[Riemannian metric]] <math>g</math> on <math>M</math>, viz a smooth association of a positive definite symmetric bilinear form <math>g_p</math> for the tangent space <math>T_p(M)</math>.
 
# A [[Riemannian metric]] <math>g</math> on <math>M</math>, viz a smooth association of a positive definite symmetric bilinear form <math>g_p</math> for the tangent space <math>T_p(M)</math>.
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==Relation with other structures==
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===Stronger structures===
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* [[Oriented Riemannian manifold]]
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===Weaker structures===
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* [[Pseudo-Riemannian manifold]]
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* [[Metric space]] {{further|[[Riemannian manifold is metric space]]}}

Latest revision as of 20:07, 18 May 2008

This article defines a differential manifold with the following additional structure -- the structure group is reduced to: orthogonal group

Definition

A Riemannian manifold is the following data:

  1. A differential manifold M
  2. A Riemannian metric g on M, viz a smooth association of a positive definite symmetric bilinear form g_p for the tangent space T_p(M).

Relation with other structures

Stronger structures

Weaker structures