Riemannian metric

From Diffgeom
Jump to: navigation, search

This article defines a tensor (viz a section on a tensor bundle over the manifold) of type (0,2)

Definition

Given data

A differential manifold M.

Definition part

A Riemannian metric g on M associates to every point p \in M a positive definite symmetric bilinear form g_p, such that g_p varies smoothly with p.

Tensoriality

A Riemannian metric is a (0,2)-tensor, viz a section of the (0,2)-tensor bundle. In fact, it is a section of the bundle Sym^2(TM), viz the bundle of symmetric 2-tensors.

Flow

Given a differential manifold, there could be many Riemannian metrics on it. A flow of a metric on the differential manifold is a Riemannian metric that gradually evolves with time.