Let $M$ be a differentiable manifold, $TM$ and $T^*M$ a tangent and cotangent bundle of $M$ and let $\Gamma (TM),\ \Gamma (T^*M)$ be spaces of smooth sections of $TM$ and $T^*M$. Let $T_s^r (M)$ denote a space of smooth tensor fields on $M$.

Since vector field on $M$ is a smooth section of $TM$, clearly

$$T_0^1 (M)=\Gamma (TM)$$ and similarly
$$T_1^0 (M)=\Gamma (T^*M).$$

My question is related to something I saw in some lecture notes on differential geometry.
There was a definition of a tensor field as a smooth section of
$TM \otimes \dots \otimes TM \otimes T^*M \otimes \dots \otimes T^*M$. But since these bundles are not vector spaces, it seems a little suspicious to me. I think the product of (co)tangent bundles may be DEFINED "fiber-wise" in the following sense

$$ TM \otimes TM \equiv \bigcup_{p \in M} \{T_pM \otimes T_pM\} \ \tag{*}$$
(disjoint union). This would also correspond with the definition of tensor field I know. Since tensor field of some type is a map which assigns to every point $p \in M$ a tensor from $T_pM \otimes \dots \otimes T_pM \otimes T_p^*M \otimes \dots \otimes T_p^*M$ smoothly,
one has

$$ T_s^r (M) = \Gamma \left( \bigcup_{p \in M} \{T_pM \otimes \dots \otimes T_pM \otimes T_p^*M \otimes \dots \otimes T_p^*M\}\right).$$

So my question is whether the following equalities are true:

$$ \bigotimes^r TM \otimes \bigotimes^s T^*M = \bigcup_{p \in M} \{\bigotimes^r T_pM \otimes \bigotimes^s T_p^*M\}\ \tag{1}$$

$$\Gamma \left( \bigotimes^r TM \otimes \bigotimes^s T^*M \right) =
\bigotimes^r \Gamma (TM) \otimes \bigotimes^s \Gamma (T^*M) \ \tag{2}$$

and if the tensor product of (co)tangent bundles is not defined the way I proposed in $(*)$, what is the correct definition or from which more general concept it follows?

This post imported from StackExchange Physics at 2014-04-01 17:32 (UCT), posted by SE-user AlanHarper