# Riemannian metric

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.