# Difference between revisions of "Riemannian manifold"

From Diffgeom

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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]] |

## Revision as of 10:41, 8 March 2007

*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 .