Space of metrics on a bundle

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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.


Gauge group acts on the space of metrics

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