Metric bundle
From Diffgeom
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 .