In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed, except for the special case of the exterior derivative of antisymmetric tensors.

Definition

A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space of the base space, which is a manifold. As such, the fiber is a vector space and the tensor bundle is a special kind of vector bundle, a tensor bundle is the fiber bundle

where V is the tangent bundle of M and V∗ is the cotangent bundle of M, and its fibers are

is the tangent space of x is the cotangent space of x, for every x in M.

References

See also