Question: To what extent do the terminology and concepts surrounding the term "tensor" (particularly as used in engineering and more applied physics, e.g. ignoring Fock spaces in QM) rely on or assume the existence of a background Riemannian metric?

Do these concepts only extend to the level of generality of Riemannian geometry, or do they use the full level of generality in which the term "tensor" can be used in abstract algebra?

Background/Motivation: When one refers to "vectors", one means either (1) the concept of "arrows" in $\mathbb{R}^3$ or perhaps more generally some Euclidean space, or (2) elements of an arbitrary abstract vector space (or even an arbitrary module), "arrows" being a specific example thereof.

A similar dichotomy also seems to exist with regards to the term "tensor", mainly between a fairly concrete viewpoint often found in physics and engineering, contrasted with the abstraction found when discussing the universal property of tensor products of modules.

Recently, from watching several lectures on YouTube, but this one in particular by Professor Grinfeld, I have come to the tnetative conclusion that concepts relating to tensors, to the level of generality found frequently in physics and engineering, e.g. upper and lower indices, Kronecker delta, Einstein summation convention, Levi-Civita symbol, all assume that some Riemannian manifold is in the background, i.e. that "tensor" refers specifically to those constructions related to the Riemannian geometry of spaces derived from $\mathbb{R}^n$.

This is because the whole notion of "index juggling" as discussed in the linked-to lecture requires the presence/use of a Riemannian metric tensor in the background. In particular, the existence of a Riemannian metric does not seem to fully capture the full level of generality which is captured by the notion of a tensor algebra of an arbitrary vector space or module.

Also the whole notion of defining a tensor as "an object which transforms in certain ways under changes of coordinates" requires the notions of "(possibly non-linear) coordinate systems" and "transformations between such possibly non-linear coordinate systems" to be defined in the background, and if I'm not mistaken, the notion of Riemannian metric which varies along the points of a Riemannian manifold is what is needed to do this.

In other words, it seems like the "engineering notion" of tensor is strictly less general than the algebraic notion, the same way that arrows in $\mathbb{R}^3$ are just a special case of algebraic vectors, but nevertheless some people still believe that they are the only kind of vector which exists.

Note: Let me know if this should better be posted on Physics.SE -- my worry is that there not enough people would be familiar with the notions of Riemannian metric or of tensor algebra to understand what exactly I am trying to ask.

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    $\begingroup$ I'm not aware of a context in physics or engineering in which tensors are used without a Riemannian metric, but it is surely the case that you can use tensors with mere differentiable manifolds (ones that do not have any sort of metric defined on them)--and lose the ability to raise or lower indices, but you still have tensors that can be defined. Is that the idea you're getting at? $\endgroup$ – Muphrid Aug 29 '16 at 1:42
  • $\begingroup$ @Muphrid Yes, this does sound right to me, because to define transformations between different curvilinear coordinate systems you just need a notion of Jacobian, and that encompasses differentiable manifolds as well (*I think). Just to confirm, the Einstein summation convention doesn't depend on the ability to raise and lower indices, correct? I think the other examples I mentioned don't depend on that either, ironically enough. But yeah engineers seem to talk about moving indices a lot in addition to moving between coordinate systems. $\endgroup$ – Chill2Macht Aug 29 '16 at 2:42
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    $\begingroup$ Yeah, summation by itself doesn't need raising or lowering of indices. $\endgroup$ – Muphrid Aug 29 '16 at 3:09

There are tensors that do not necessarily come from a Riemannian metric and still play an important role in applications. For instance in physics, in the field of classical field theory (and probably in quantum field theory too), a (classical) field is a connection defined on an appropriate principal or vector bundle over space-time. The curvature of that connection is the strength of the field (it basically represents the force with which the field acts) and this curvature is a tensor field. For instance the electromagnetic field is a tensor field corresponding to the curvature of the line bundle connection given by the electromagnetic vector potential (observe, the vector potential is not a vector field, it is a connection one-form). Also, in the case of Poisson manifolds, which for instance are a general framework even for classical mechanics, you have something called a Poisson bracket, which is represented by an antisymmetric tensor field which is usually referred to as a bi-vector. In the more special case of symplectic geometry and symplectic manifolds (again, this is the traditional framework for classical mechanics) you have a non-degenerate closed two-form, which is also an antisymmetric tensor field. Just like the Riemanian metric, you use it to raise and lower indices, i.e. to move back and forth between tangent vectors and dual (cotangent) vectors. Probably there are many other examples, but these ones I came up with just now.

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  • $\begingroup$ Just to confirm, all of these tensors are defined on differentiable manifolds, correct? So we can understand them as "objects which change a certain way under changes of coordinates"? Wikipedia says that both Poisson and symplectic manifolds are both smooth manifolds, but I just wanted to confirm. Also, for symplectic manifolds, is the raising and lowering of indices accomplished using the symplectic bilinear form? Which I guess is a similar but still significantly different object from a symmetric bilinear form (i.e. quadratic form). $\endgroup$ – Chill2Macht Aug 29 '16 at 15:57
  • $\begingroup$ Exactly. These objects change as tensors under change of coordinates. This is because they can be intrinsically defined on the manifold without the use of coordinates. They are all objects living on smooth manifolds. Of course in many applications they arise in coordinate form (because we choose a coordinate system in which we describe our models). I think it is interesting and instructive to understand electromagnetic theory as a connection whose curvature is the electromagnetic field. $\endgroup$ – Futurologist Aug 29 '16 at 17:23
  • $\begingroup$ The rising and lowering of indices in symplectic geometry, with respect to the symplectic form is analogous to the rising and lowering of indices in Riemannian geometry. Both the metric and the symplectic form establish an isomorphism between the tangent and the cotangent bundles. This is exactly the mechanism behind raising and lowering indices.That's why the symplectic form needs to be non-degenerate. $\endgroup$ – Futurologist Aug 29 '16 at 17:25

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