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.