Metric bundle

Definition

Standard definition

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

• A vector bundle $E$ over $M$
• For every point $p\in M$, a symmetric positive-definite bilinear form on the vector space $E(p)$ over $p$, that varies smoothly with $p$.

Definition as a section

A metric on $E$ is defined as a section of the bundle $Sym^{2}(E^{*})$, with the property that the value of the section at every point is positive-definite. Note that a section of $Sym^{2}(E^{*})$ is precisely the same thing as associating, to every point of $M$, 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 $Sym^{2}(E^{*})$.