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.

 

[tacman]

Yes, there is a relationship between tensors used in differential geometry and the notion of tensor product as used in module theory. In fact, tensors in differential geometry can be viewed as a special case of tensor products in module theory.

In differential geometry, tensors are used to represent geometric objects such as vectors, covectors, and higher order tensors in a coordinate-independent manner. They are defined as multilinear maps that take in vector and covector inputs and produce a scalar output. These tensors can be manipulated and transformed using coordinate transformations without changing their intrinsic geometric properties.

On the other hand, tensor products in module theory are used to combine two modules into a new module. Just like tensors in differential geometry, tensor products are also multilinear maps. However, in this context, they take in elements from two different modules and produce a new element in the tensor product module.

The relationship between these two concepts lies in the fact that tensors in differential geometry can be seen as a special case of tensor products in module theory. In particular, the space of tensors on a manifold can be identified with the tensor product of the tangent and cotangent spaces at each point on the manifold. This allows us to use the machinery of module theory to manipulate tensors in differential geometry.

To summarize, tensors in differential geometry and tensor products in module theory are related through the concept of multilinear maps. Tensors in differential geometry can be seen as a special case of tensor products, which allows for a deeper understanding and manipulation of these geometric objects.