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A reference for this material is Chapter 12 of John M. Lee. Introduction to smooth
manifolds. Second edition. Grad. Texts in Math., 218. Springer, New York, 2013.
xvi+708pp. ISBN: 978-1-4419-9981-8.
The \((k,\ell )\)-tensor bundle over \(M\) is defined as
In other words, a tensor of type \((\ell , k)\) can be regarded as
a multilinear function that takes \(k\) vector fields and returns a linear combiation of
tensor products of \(\ell \) vector fields.
Let \( F : \mathfrak {X} (M) \times \ldots \times \mathfrak {X} (M) \to C^{\infty } (M) \) be a function that takes \(k\) vector fields and returns a function. It is a \(C^{\infty }(M)\)-module
morphism if it is multilinear and for any \(h \in C^{\infty } (M) \) and any vector fields \(X_1, \ldots , X_k \in \mathfrak {X}(M)\), one has
\begin{align*} F (h X _1, \ldots , X_k) & = F(X_1, hX_2, \ldots , X_k) \\ & = \ldots \\ & = F(X_1, \ldots , hX_k ) \\ & = h F (X_1, \ldots , X_k) . \end{align*}
By the above discussion, this happens if and only if \(F\) is associated to a tensor field \(\tilde {F} \in \mathcal {T}^k_0(M)\).
A more direct way of seeing this is the following: for finite-dimensional vector spaces,
we have a natural isomorphism
\[ V^{\ast } \otimes W \cong Hom(V , W) . \]
Iterating this,
\begin{align*} V ^{\ast } \otimes \cdots \otimes V^{\ast } & \cong Hom ( V , V^{\ast } \otimes \cdots \otimes V^{\ast } ) \\ & \cong Hom ( V , Hom ( V , V ^{\ast } \otimes \cdots \otimes V^{\ast } ) ) \\ & \cong Hom ( V , Hom ( V , Hom (V , V ^{\ast } \otimes \cdots \otimes V^{\ast } ) ) ) \\ & \ldots \\ & \cong Hom (V , Hom (V, \ldots , Hom (V, \mathbb {R}) \cdots )) \\ & \cong Mult_{k} (V , \ldots , V ; \mathbb {R} ) \\ & \cong ( V \otimes \cdots \otimes V )^{\ast }, \end{align*}
where the last isomorphism comes from the fact that linear maps
\[ V \otimes \cdots \otimes V \to \mathbb {R} \]
are in one-to-one
correspondence with \(k\)-multilinear maps
\[ V \times \cdots \times V \to \mathbb {R}. \]
This show that there is a natural
isomorphism
\[ V ^{\ast } \otimes \cdots \otimes V^{\ast } \cong ( V \otimes \cdots \otimes V )^{\ast } . \]
These isomorphisms pass to vector bundles, so for any vector bundle \(E \to M \)
we have
Smooth manifolds