On the definition of a tensor in the study of manifolds

Question

In the book An introduction to manifolds by Tu, Loring W, it defines the $k$-tensor as follows

Denote by $V^{k}=V \times \cdots \times V$ the Cartesian product of $k$ copies of a real vector space $V $. A function $f: V^{k} \rightarrow \mathbb{R}$ is $ k $-linear if it is linear in each of its $ k $ arguments: $$ f(\ldots, a v+b w, \ldots)=a f(\ldots, v, \ldots)+b f(\ldots, w, \ldots) $$ for all $ a, b \in \mathbb{R} $ and $ v, w \in V $. Instead of $2$-linear and $3 $-linear, it is customary to say "bilinear" and "trilinear." A $k $-linear function on $V$ is also called a $ k $-tensor on $ V $. We will denote the vector space of all $ k $-tensors on $ V $ by $ L_{k}(V) $. If $f $ is a $ k $ -tensor on $ V $, we also call $ k $ the degree of $ f $.

However in Wikipedia, it defines the $(p,q)$-tensor as follows

In this approach, a type $(p,q)$ tensor $T$ is defined as a multilinear map, $$ T: \underbrace{ V^* \times\dots\times V^*}_{p \text{ copies}} \times \underbrace{ V \times\dots\times V}_{q \text{ copies}} \rightarrow \mathbf{R}, $$ where $V^*$ is the corresponding dual space of covectors, which is linear in each of its arguments.

It is obvious that the definition of a $k$-tensor in An introduction to manifolds is just the $(0,q)$-tensors in the definition of Wikipedia.

My question is, why do not we define the tensors to be $(p,q)$-tensor in Tu, Loring W?

EDIT: I have noticed that there are no $(p,q)$-tensors in the whole book of Loring Tu.

Is it because that the elements from dual space will never be used in tensors in manifolds in the following study (so $p$ always equals to $0$)? If not, where do we use $(p,q)$-tensors in the study of manifolds? (maybe it will appear in tensor fields?)

I would really appreciate it if anyone could tell me the connections of these things.

Answer 1

Note that even Wikipedia's definition is not the most general. The most general definition (that I know of) for a (real) tensor is a multilinear map $$ T: V_1 \times V_2 \times \cdots \times V_k \rightarrow \mathbb{R},$$ where $V_1, \dots, V_k$ are (not necessarily the same) vector spaces. And such generalized tensors definitely do appear in differential geometry and topology.

I also want to note that these simple definitions are also very misleading in the sense that the tensors that arise in geometry and topology all have symmetries or anti-symmetries in the indices. This leads to most of the headaches when learning how to deal with tensors. In fact, if you get serious with tensors, the notation can become quite a challenge. Penrose developed a particularly creative approach to dealing with this.

However, when you're first learning differential geometry and topology, almost all of the tensors encountered are indeed $(0,q)$ tensors. That's because either you're working with only exterior differential forms or you're working with a Riemannian manifold, where the metric can be used to convert any $(p,q)$-tensor into $(0,p+q)$-tensor.

It's good to know the definitions of general tensors (as described above) and $(p,q)$ tensors, because they are sometimes necessary or useful. But it's probably best to avoid devoting much attention to them in introductory courses and textbooks.

ADDED: An extremely useful canonical isomorphism is $$ \operatorname{Hom}(V,W) = V^*\otimes W $$ where $V$ and $W$ are vector spaces and $\operatorname{Hom}(V,W)$ is the space of linear maps from $V$ to $W$. Used with care, this means that you can view a $(p,q)$ tensor $T \in V_1^*\otimes\cdots\otimes V_q^*\otimes V_{q+1} \otimes \cdots \otimes V_{p+q}$ as a multilinear map $$ T: V_{1}\times\cdots\times V_{q} \rightarrow V_{q+1}\otimes\cdots\otimes V_{p+q}. $$ In short, to avoid working with $(p,q)$-tensors directly, it can be useful to view them as maps from $(q,0)$-tensors to $(p,0)$-tensors instead. Or, equivalently, as $(V_{q+1}\otimes\cdots\otimes V_{p+q})$-valued $(0,q)$-tensors.