Metric tensor
Metric as a section of a bundle

By the universal property of the tensor product, any bilinear mapping (10) gives rise naturally to a section g of the dual of the tensor product bundle of T M with itself

The section g is defined on simple elements of T M ⊗ T M by

and is defined on arbitrary elements of T M ⊗ T M by extending linearly to linear combinations of simple elements. The original bilinear form g is symmetric if and only if

where

is the braiding map.

Since M is finite-dimensional, there is a natural isomorphism

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