Definition of tensors

Question

I'm studying the book "Riemannian Geometry" by Petersen and since I'm new to the subject, I'm helping myself also with the more introductory DoCarmos's book. I'm a bit confused about the definition of tensor. The definition given by DoCarmo is the following.

Given a Riemannian manifold M, a tensor $T$ of order $r$ is a multilinear map $$ T: \underbrace{\mathcal{X}(M) \times \dots \times \mathcal{X}(M)}_{r \text{ times}} \to \mathcal{D}(M)$$ where $\mathcal{X}(M) $ is the set of all the smooth vector fields over $M$ and $\mathcal{D}(M)$ is the set of all the smooth functions on $M$.

But Petersen's book talks about $(q, r)$-tensors and I guessed from the context that that for him a $(q, r)$-tensor is a multilinear map

$$ T: \underbrace{\mathcal{X}(M) \times \dots \times \mathcal{X}(M)}_{r \text{ times}} \to \underbrace{\mathcal{X}(M) \times \dots \times \mathcal{X}(M)}_{q \text{ times}}$$.

Then I took a look also to John Lee's book and I found this definition.

A $r$-covariant and $q$-contravariant tensor on a real vector space $V$ is a multilinear map $$T: \underbrace{V^* \times \dots \times V^*}_{q \text{times}} \times \underbrace{V \times \dots \times V} \to \mathbb{R}. $$

Then he extend the definition to manifolds considering smooth sections of tensor bundles.

What is the differences between these approaches? Could someone explain me better how Petersen sees tensors?

Answer 1

Given a $q$-contravariant $r$-covariant tensor in the sense of John Lee's book, You can get a different multilinear map from $q-1$ copies of $V^*$ and $r$ copies of $V$ to $V$ as follows.

Let $\bar{T}(\omega_2,\ldots,\omega_q,v_1,\ldots,v_r)$ be an element $w\in V$ such that $\omega_1(w) = T(\omega_1,\ldots,\omega_q,v_1,\ldots,v_r)$ for any $\omega_1$. Repeating this, we can get a map from $q-2$ copies of $V^*$ and $r$ copies of $V$ to $2$ copies of $V$, and so on until we get a multilinear map from $r$ copies of $V$ to $q$ copies of $V$. This gives a (natural) isomorphism between $q$-contravariant $r$-covariant tensors to multilinear maps from $q$ copies of $V$ to $r$ copies of $V$.

That's the "tricky" part. Extending that from vectors and scalars to vector fields and functions gets you to Petersen's definition. That's a little hand-wavy, but hopefully puts you on the right track.

Edit: By the way, just picked up my copy of Lee's Riemmannian Manifolds book ( great book, second edition coming in 2017 according to rumour, can't wait ), and what I wrote above is basically Lemma 2.1