# Questions on notation that I haven't seen before (tensor product sign in superscript and mysterious $\Gamma$ symbol)

I have some questions about the Wikipedia article on Tensor fields. The definition of a tensor field is given as such:

Let $$\mathfrak M$$ be a manifold, for instance the Euclidean plane $$\mathbb R^n$$.

DEFINITION: A tensor field of type $$(p, q)$$ is a section

\begin{align} T \in \Gamma( \mathfrak M , V^{\otimes p} \otimes (V^*) ^{\otimes q}) \end{align} where v is the vector bundle on $$\mathfrak M$$, $$V^*$$ is its dual and $$\otimes$$ is the tensor product of vector bundles.

(a)

Is the $$^{\otimes p,q}$$ analogous to the $$^n$$ in cartesian product $$\mathbb R ^n$$? That is, we take the tensor product of $$V$$ with itself $$p, q$$ times? Concretely, would writing $$V^{\otimes 3}$$ expand to $$V \otimes V \otimes V$$.

(b)

The editor gave some definitions, but neglected to write what $$\Gamma$$ is intended to mean. I haven't seen $$\Gamma$$ used in this context before, and I have read some books which involve manifolds and vector bundles. I would highly appreciate someone telling me what it is meant to represent.

Thank you in advance for help with both of these questions.

• (a)- correct. (b) $\Gamma$ denotes the space of sections of a bundle. Jan 28 at 2:40
• @MoisheKohan ty
– Nate
Jan 28 at 3:29

(b) If $$\xi: E\to B$$ is a bundle (a smooth vector bundle in your case) then $$\Gamma(\xi)$$ or, sometimes, $$\Gamma(B, E)$$, denotes the space of sections of this bundle (in your case, the space of smooth sections).