Riemannian manifold: Difference between revisions
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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>. | ||
==Relation with other structures== | |||
===Stronger structures=== | |||
* [[Oriented Riemannian manifold]] | |||
===Weaker structures=== | |||
* [[Pseudo-Riemannian manifold]] | |||
* [[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:
- A differential manifold
- A Riemannian metric on , viz a smooth association of a positive definite symmetric bilinear form for the tangent space .
Relation with other structures
Stronger structures
Weaker structures
- Pseudo-Riemannian manifold
- Metric space Further information: Riemannian manifold is metric space