I'm reading Lee's Introduction to Smooth Manifolds, and in Chapter 12 I've hit a minor but annoying wrinkle regarding tensor fields. Proposition 12.25(a) states that for any two covariant tensor fields $A$ and $B$ and smooth map $F : M \to N$,
$F^* (A \otimes B) = F^* A \otimes F^* B$
where $F^* A$ is the pullback of $A$ by $F$. Proving the proposition is straightforward enough, but in doing so I make use of $(A \otimes B)_p = A_p \otimes B_p$, which seems intuitively obvious and is actually the definition of the tensor product of tensor fields in another text. But unfortunately, Lee never explicitly states this, and it seemingly can't be derived from any other information in the chapter. In fact, after defining a covariant tensor field on a smooth manifold $M$ as a section of the tensor bundle
$$ T^k T^* M = \coprod_{p \in M} T^k(T^*_p M), $$
(where $T^k(T^*_p M)$ is the vector space of covariant $k$-tensors on $T_p M$) Lee simply states that any such section $A$ can be written (using the summation convention) as
$$ A = A_{i_1 \dots i_k} dx^{i_1} \otimes \dots \otimes dx^{i_k}, $$
without any mention of how one is to evaluate the given tensor product of covector fields at a point.
Is there any way I can arrive at the desired result using strictly what is presented by Lee, or am I being overly pedantic and should just move on?
Best Answer
I'm surprised by this question, because my writing is more often criticized for being too pedantic! Did you really have to wonder what $A\otimes B$ means? I'm having a hard time imagining what it could possibly mean other than $(A\otimes B)_p = A_p\otimes B_p$. But you're right, I didn't actually write that definition anywhere. To forestall any future confusion, I've added it to my correction list.