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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.

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

1 Answer 1

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To convert my comment to a proper answer.

(a) Your understanding is correct.

(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).

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