This article defines a tensor (viz a section on a tensor bundle over the manifold) of type (0,2)
A Riemannian metric on associates to every point a positive definite symmetric bilinear form , such that varies smoothly with .
A Riemannian metric is a -tensor, viz a section of the -tensor bundle. In fact, it is a section of the bundle , viz the bundle of symmetric 2-tensors.
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.