By the universal property of the tensor product, any bilinear mapping () gives rise naturally to a section of the dual of the tensor product bundle of with itself

The section is defined on simple elements of by

and is defined on arbitrary elements of by extending linearly to linear combinations of simple elements. The original bilinear form is symmetric if and only if
where
is the braiding map.

Since is finite-dimensional, there is a natural isomorphism

so that is regarded also as a section of the bundle of the cotangent bundle with itself. Since is symmetric as a bilinear mapping, it follows that is a symmetric tensor.

Metric as a section of a bundle
Examples
Area
Canonical measure and volume form