# Metric bundle

## Contents

## Definition

### Standard definition

Let be a differential manifold. A **metric bundle** on is the following data:

- A vector bundle over
- For every point , a symmetric positive-definite bilinear form on the vector space over , that varies smoothly with .

### Definition as a section

A metric on is defined as a section of the bundle , with the property that the value of the section at every point is positive-definite. Note that a section of is *precisely* the same thing as associating, to every point of , a symmetric bilinear form. The condition of positive-definiteness needs to be imposed additionally to get a metric.

## Related notions

### Space of metrics on a bundle

`Further information: space of metrics on a bundle`

Given any vector bundle over a differential manifold, we can look at the space of metrics on it. This space lives as a subset (*not* a vector subspace) of the space of sections of .