Relationship between tensors

[sparkster]

Is there any relationship between tensors, as they're used in diff geo and the notion of tensor product as used in module theory? I seem to recall that tensor products were "invented" because, given a field k and U, V two vector spaces over k such that dim U=n, dim V=m, we wanted to construct a vector space with dimension nm.

But I'm not sure where tensors used the other way came from, thus my question.

 

[river_rat]

Yes there is, the tensor fields as used in differential geometry are constructed by taking sections of the tensor product of copies of the tangent bundle and cotangent bundle. The tensor product is also important in that it is used as a starting point to define the exterior product of covectors.