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I'm having a hard time understanding what's going on with tensor fields.

I understand that $A$ is a smooth covariant tensor field of order $s$ (or a $(0,s)$ tensor field) on a smooth manifold $M$ if it is simply an $\mathbb{R}$-multilinear map from $s$ products of $\Gamma(TM)$ into the space of all the smooth scalar-valued functions on $M$ such that \begin{equation} A(f_1X_1,\dots,f_sX_s)(p)=f_1(p)\dots f_s(p)A(X_1,\dots,X_s)(p) \end{equation} for all $f_1,\dots,f_s$ smooth and all $X_1,\dots,X_s$ smooth vector fields and all $p$.

Now here comes my confusion and question. Can I think of a $(1,s)$ smooth tensor field on $M$ as an $\mathbb{R}$-multilinear map from $s$ products of $\Gamma(TM)$ into $\Gamma(TM)$ which satisfies the same equation as above? Is this the right way of thinking about a $(1,s)$ tensor field?

If that's true, how can I think of a $(2,s)$ tensor field on $M$?

Thank you.

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Yes, you're right, and for the $(2,s)$ case you'll end up in $\Gamma(TM\otimes TM)$.

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  • $\begingroup$ Thanks. I think I get it now. $\endgroup$
    – Axiom
    Feb 20, 2015 at 2:48

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