# 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== | ||

+ | |||

+ | ===Stronger structures=== | ||

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+ | * [[Oriented Riemannian manifold]] | ||

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+ | ===Weaker structures=== | ||

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+ | * [[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

## Contents

## 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`