Space of metrics on a bundle

Definition

Let $M$ be a differential manifold and $E$ be a vector bundle. The space of metrics on $E$ is the set of all possible ways of giving $E$ the structure of a metric bundle.

This can be viewed as a subset of the space of sections of $Sym^{2}(E^{*})$.

In the particular case where $E=TM$, we get the space of Riemannian metrics.

Facts

Gauge group acts on the space of metrics

The gauge group of the vector bundle $E$ acts on the space of metrics of $E$.