What is the modern approach to tensors?

Question

I have recently started studying tensors a bit from the index notation point of view -- I understand contractions and the metric tensor and such well enough.

However I've been told that this approach to tensors (the physicist's approach) is old-fashioned. So what is the modern mathematical approach? Does it have to do with tensor products? Or multilinear functions? Or whatever vector bundles and coherent sheaves are? (I saw each of these on the Wikipedia page.)

Is there an introductory book that goes over the modern approach?

Answer 1

I recommend Lee's Riemannian Geometry: An Introduction to Curvature. As the name suggests, it is a book about about Riemannian Geometry but chapter $2$ is titled 'Review of Tensors, Manifolds, and Vector Bundles'. I remember reading it when I was coming to grips with tensors and I found it very clear.

There are two things which are often called tensors which Lee's text distinguishes between. One is a tensor on a vector space, while the other is a tensor on a manifold. The latter is what Lee (and subsequently I) calls a tensor field; this is completely analogous to the difference between a vector and a vector field. After first learning what tensors on a vector space are, you can then move on to tensor fields which you can think of as smoothly varying tensors - once you're at this point, you may want to learn about their description as sections of certain vector bundles called tensor bundles.

Following Lee, a $(k, l)$-tensor on a real vector space $V$ is a multilinear map $F : (V^*)^l\times V^k \to \mathbb{R}$. Once you choose a basis $\{E_i\}$ for $V$, you obtain a collection of real numbers $F_{i_1\dots i_k}^{j_1,\dots j_l}$ (the coefficients in a linear combination). When moving on to tensor fields, the $\{E_i\}$ become a (local) basis of vector fields and the coefficients $F_{i_1\dots i_k}^{j_1,\dots j_l}$ become smooth real-valued functions. This collection of functions (or just the symbols which represent them) is usually what physicists use when dealing with tensors (or more precisely, tensor fields).

Answer 2

I love Expository papers by Keith Conrad. He has two papers on tensor products here are the links- 1 and 2. This is the algebraist Point of view of tensors and as far as I know it is modern view too.

You can find Expository papers on various topics in algebra on his home page.

He has also written at the end of Tensor Products-$1$ that to see a physicist introduce tensors (really, tensor felds) as indexed quantities, watch Leonard Susskind's lectures (lecture 3,4 and 5) on general relativity on YouTube. If it interests you.