What does the curvature of a surface have to do with tensor products?

The question I would like to address in this article is: what is a tensor? This question has two answers. If you ask an algebraist, he (she) will tell you it is an element of the tensor product of two modules. If you ask a geometer, she (he) will ramble about “global constructions that only depend on point-wise values” for hours on end. We should note that we will primarily focus on what a tensor is from the perspective of a geometer, the intuition behind it and how we get from that to the usual formalism.

Frankly, I feel like there isn’t much to explain, yet I never had this explained to me and I always felt it was difficult reconcile my intuition with the formalism most commonly adopted. This is the primary reason I wrote this article: I would have loved to read it in the past.

In differential geometry and related fields, information can often be obtained by passing from the non-linear to the linear via infinitesimal approximations. Often times this comes in form of C-linear functions between the spaces of smooth sections of two fiber bundles. Specifically, if M is a smooth manifold and EM, FM are vector bundles over M, then the sets Γ(E) and Γ(F) of smooth (global) sections of E and F, respectively, have a natural structure of a C(M)-modules, and sometimes linear maps Γ(E)Γ(F) show up. If such a map τ:Γ(E)Γ(F) satisfies the condition that τ(ξ)p depends only on ξp — and not ξ on its entirety — then τ is called a tensor.

Often times it is convenient to also consider multilinear maps τ:Γ(E1)×Γ(E2)××Γ(En)F — i.e. maps that are linear in each coordinate. Again, if τ(ξ1,ξ2,,ξn)p is determined by ξpi then τ is called a tensor. The classical examples of tensors are differential forms. A perhaps more interesting example is a Riemannian metric: for each point pM we fix a positive-definite bilinear form gp:TpM×TpM which “varies smoothly with p”. This construction induces a tensor

g:X(M)×X(M)C(M)Γ(M×)

where (g(V1,V2))(p)=gp(Vp1,Vp2).

This is what a tensor is supposed to be: for each pM we fix some multilinear function between the p-fibers of some vector bundles that “varies smoothly with p”. The meaning of “varies smoothly with p” is still imprecise, dare I not say unclear. We should point out that often times it is more convenient to define tensors in terms of global sections rather than defining the fiber-wise transformations, such as in the case of the curvature tensor R(X,Y)Z=XYZYXZ[X,Y]Z of a connection or the Nijenhuis tensor N(X,Y)=[X,Y]+J[JX,Y]+J[X,JY][JX,JY] of an almost complex structure J.

Hence the need to consider tensors in geometry. Working with multilinear maps can be a bit of an annoyance, however. It would be convenient if we could somehow look at a tensor as a straight linear map — instead of a multilinear map. This brings us to the algebraic answer to our initial question. Given a ring R and two R-modules M and N, their tensor product MRN is the R-module which enjoys the universal property that

HomR(MRN,L)Bil(M×N,L),

where Bil(M×N,L) is the module of R-bilinear maps M×NL.

In other words, R-multilinear maps M1×M2××MnN naturally correspond to R-linear maps M1M2MnN. We should point out that the tensor product of modules can always be shown to exist by means of an explicit construction — whose elements are usually called tensors. If we fix R=, this construction induces a construction in the category of vector bundles over some fixed manifold M: if EiM are bundles over M, there is a vector bundle E1E2EnM whose fibers are

(E1E2En)|p=E1|pE2|pEn|p

The relationship between these two notions of tensor should now be clear: tensors Γ(E1)×Γ(E2)××Γ(En)Γ(F) are called tensors because they correspond to C(M)-linear maps

Γ(E1)C(M)Γ(E2)C(M)C(M)Γ(En)Γ(F),

which are in turn canonically identified with C(M)-linear maps

Γ(E1E2En)Γ(F)

In fact, there’s a natural isomorphism of sheaves of C-modules Γ(,E1)CΓ(,E2)CCΓ(,En)Γ(,E1E2En) 🤡

To recap: we’ve just shown that a tensor τ:Γ(E1)×Γ(En)Γ(F) can be naturally identified with some τHomC(M)(Γ(E1En),Γ(F)). A natural question to ask ourselves at this point is: does τ correspond to some τΓ(Hom(E1En,F))? First of all, why does this make sense? Recall that given two vector spaces V and W, the set Hom(V,W) of linear transformations VW is again a vector space. Hence we can consider the vector bundle Hom(E1En,F)M whose fibers are

Hom(E1En,F)|p=Hom(E1|pEn|p,F|p)

The previously mentioned example of Riemannian metrics does hint at an inclusion

i:Γ(Hom(E1En,F))HomC(M)(Γ(E1En),Γ(F)),

which takes ηΓ(Hom(E1En,F)) to iη:Γ(E1En)Γ(F) with iη(ξ)p=ηp(ξp) — notice this is precisely what we did to get from “a bilinear form in TpM for each pM” to a Riemannian metric seen as a tensor. The meaning of “a transformation at each fiber p that varies smoothly with p” is now much clearer too: this is a smooth section of Hom(E1En,F). The inclusion i is not surjective. This is because in general if φ:Γ(E1En)Γ(F) is a homomorphism the value of φ(ξ1,,ξn)p may very well depend on ξi in their entirety, and not only on ξpi.

This last statement is actually false! See the errata on this post.

We claim, however, that the image of i consists precisely of the multilinear functions E1××EnF that are tensors — i.e. such that τ(ξ1,,ξn)p is determined by ξpi. Indeed, if we consider the map

s:Τ(E1××En,F)Γ(Hom(E1En,F))

given by sτp(v1,,vn)=τ(ξ1,,ξn)p, where Τ(E1×En,F)HomC(M)(E1En,F) is the subspace of tensors and ξiΓ(Ei) are such that ξpi=vi, we can very quickly check that i=s1, establishing an isomorphism of C(M)-modules

Γ(Hom(E1En,F))Τ(E1××En,F)

The definition of sτp(v1,,vn) does not depend on the choice of ξi precisely because the value of τ(ξ1,,ξn)p depends only on ξpi=vi! In conclusion, a tensor τ:E1××EnF is called a tensor because it corresponds to a smooth section of Hom(E1En,F). To finish things of, I would like to conclude our discussion by explaining a small notational quirk the reader will probably encounter in the literature: most people refer to Hom(E1En,F) as E1*En*F.

This is because given two vector spaces V and W, the space Hom(V,W) is canonically isomorphic to V*W. Taking V=E1|pEn|p and W=F|p, this translates to an isomorphism of vector bundles Hom(E1En,F)E1*En*F. In fact, usually the differential structure of Hom(E1En,F) is defined via the identification with E1*En*F. This is the formalism generally adopted, which is to say, when a geometer says “a tensor” in a formal sense he most likely means “some τΓ(E1*En*F)”.

Also, if FM is the trivial line bundle M×, one usually refers to E1*En*F by simply E1*En*, because tensoring by M× is the same as doing nothing. For instance, a Riemmanian metric is most often defined as a tensor gΓ(T*MT*M) satisfying special conditions. That about wraps it up. I hope this helped someone 😛