Metric bundle

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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^*).