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

• Gavroy
In summary, the conversation discusses the use of notation involving the Minkowski space and the tensor product symbol in the context of vector spaces and manifolds. The concept of the cotangent bundle and cotangent space is also mentioned. It is suggested that TM* may be a variation on this notation and the use of the T symbol is explained.

#### Gavroy

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

[TEX]TM* \otimes TM*[/TEX]

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?

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

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 TpM the tangent space associated with point p.)

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