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 $Sy{m}^2(TM)$, 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.