Understanding TM* (tensorproduct) TM*: Let's Break it Down!

[Gavroy]

hi
can somebody explain this notation to me
Let M be the minkowski space, then there is a space:

TM* \otimes TM*

For some reason, this latex code does not work:

TM* (tensorproduct) TM*

I don't really get what this T tells me?

Can somebody explain this to me?

 

[brydustin]

This has nothing to do with vector spaces, linear algebra, or abstract algebra. This post should be removed.

 

[Rasalhague]

I've seen T^*M used to mean the cotangent bundle of a manifold, and T^*_pM the cotangent space associated with point p \in M (e.g. Lee: Riemannian Manifolds, p. 17), where M is the underlying set of the manifold. The use of the tensor product symbol in this context is explained here:

http://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces

In this context, I would read T^*_P M \otimes T^*_p M to mean the vector space whose vectors are tensors that can be written in the form

\sum_{\mu = 0}^3 \sum_{\nu=0}^3 \omega_{\mu\nu} \mathbf{e}^\mu \otimes \mathbf{e}^\nu

where \omega_{\mu\nu}, for all possible values of mu and nu, are the coefficients, a.k.a. (scalar) components, and \mathbf{e}^\mu, for all possible values of mu (or nu, as the case may be), are basis covectors, i.e. basis vectors of the dual space to the tangent space associated with point p of, in this case, Minkowski space.

My first guess would be that your TM^* is a variation on this notation. Does that seem likely from the context? (With no star, I'd read TM as the tangent bundle on M, and T_pM the tangent space associated with point p.)