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 β€œglobal 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∞C^\infty-linear functions between the spaces of smooth sections of two fiber bundles. Specifically, if MM is a smooth manifold and Eβ†’ME \rightarrow M, Fβ†’MF \rightarrow M are vector bundles over MM, then the sets Ξ“(E)\Gamma ( E ) and Ξ“(F)\Gamma ( F ) of smooth (global) sections of EE and FF, respectively, have a natural structure of a C∞(M)C^\infty ( M )-modules, and sometimes linear maps Ξ“(E)β†’Ξ“(F)\Gamma ( E ) \rightarrow \Gamma ( F ) show up. If such a map Ο„:Ξ“(E)β†’Ξ“(F)\tau : \Gamma ( E ) \rightarrow \Gamma ( F ) satisfies the condition that Ο„(ΞΎ)p\tau {\left ( \xi \right )}_p depends only on ΞΎp\xi_p — and not ΞΎ\xi on its entirety — then Ο„\tau is called a tensor.

Often times it is convenient to also consider multilinear maps Ο„:Ξ“(E1)Γ—Ξ“(E2)Γ—β‹―Γ—Ξ“(En)β†’F\tau : \Gamma ( E_1 ) \times \Gamma ( E_2 ) \times \cdots \times \Gamma ( E_n ) \rightarrow F — i.e. maps that are linear in each coordinate. Again, if Ο„(ΞΎ1,ΞΎ2,…,ΞΎn)p\tau(\xi^1, \xi^2, \ldots, \xi^n)_p is determined by ΞΎpi\xi_p^i then Ο„\tau is called a tensor. The classical examples of tensors are differential forms. A perhaps more interesting example is a Riemannian metric: for each point p∈Mp \in M we fix a positive-definite bilinear form gp:TpMΓ—TpMβ†’R\text{g}_p : T_p M \times T_p M \rightarrow \mathbb{R} which β€œvaries smoothly with pp”. This construction induces a tensor

g:X(M)Γ—X(M)β†’C∞(M)β‰…Ξ“(MΓ—R)\mathrm{g} : \mathfrak{X}(M) \times \mathfrak{X}(M) \to C^\infty(M) \cong \Gamma(M \times \mathbb{R})

where (g(V1,V2))(p)=gp(Vp1,Vp2)\left ( \text{g} ( V^1 , V^2 ) \right ) ( p ) = \text{g}_p ( V_p^1 , V_p^2 ).

This is what a tensor is supposed to be: for each p∈Mp \in M we fix some multilinear function between the pp-fibers of some vector bundles that β€œvaries smoothly with pp”. The meaning of β€œvaries smoothly with pp” 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=βˆ‡Xβˆ‡YZβˆ’βˆ‡Yβˆ‡XZβˆ’βˆ‡[X,Y]ZR ( X , Y ) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[ X , Y ]} Z of a connection βˆ‡\nabla or the Nijenhuis tensor N(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]βˆ’[JX,JY]N ( X , Y ) = [ X , Y ] + J [ J X , Y ] + J [ X , J Y ] - [ J X , J Y ] of an almost complex structure JJ.

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 RR and two RR-modules MM and NN, their tensor product MβŠ—RNM \otimes_R N is the RR-module which enjoys the universal property that

HomR(MβŠ—RN,L)β‰…Bil(MΓ—N,L),\text{Hom}_R \left ( M \otimes_R N , L \right ) \cong \text{Bil} \left ( M \times N , L \right ) ,

where Bil(M×N,L)\text{Bil} \left ( M \times N , L \right ) is the module of RR-bilinear maps M×N→LM \times N \rightarrow L.

In other words, RR-multilinear maps M1Γ—M2Γ—β‹―Γ—Mnβ†’NM_1 \times M_2 \times \cdots \times M_n \rightarrow N naturally correspond to RR-linear maps M1βŠ—M2βŠ—β‹―βŠ—Mnβ†’NM_1 \otimes M_2 \otimes \cdots \otimes M_n \rightarrow N. 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=RR = \mathbb{R}, this construction induces a construction in the category of vector bundles over some fixed manifold MM: if Eiβ†’ME_i \rightarrow M are bundles over MM, there is a vector bundle E1βŠ—E2βŠ—β‹―βŠ—Enβ†’ME_1 \otimes E_2 \otimes \cdots \otimes E_n \rightarrow M whose fibers are

(E1βŠ—E2βŠ—β‹―βŠ—En)|p=E1|pβŠ—E2|pβŠ—β‹―βŠ—En|p\left ( E_1 \otimes E_2 \otimes \cdots \otimes E_n \right ) \text{|}_p = E_1 \text{|}_p \otimes E_2 \text{|}_p \otimes \cdots \otimes E_n \text{|}_p

The relationship between these two notions of tensor should now be clear: tensors Ξ“(E1)Γ—Ξ“(E2)Γ—β‹―Γ—Ξ“(En)β†’Ξ“(F)\Gamma ( E_1 ) \times \Gamma ( E_2 ) \times \cdots \times \Gamma ( E_n ) \rightarrow \Gamma ( F ) are called tensors because they correspond to C∞(M)C^\infty ( M )-linear maps

Ξ“(E1)βŠ—C∞(M)Ξ“(E2)βŠ—C∞(M)β‹―βŠ—C∞(M)Ξ“(En)β†’Ξ“(F),\Gamma ( E_1 ) \otimes_{C^\infty ( M )} \Gamma ( E_2 ) \otimes_{C^\infty ( M )} \cdots \otimes_{C^\infty ( M )} \Gamma ( E_n ) \rightarrow \Gamma ( F ) ,

which are in turn canonically identified with C∞(M)C^\infty ( M )-linear maps

Ξ“(E1βŠ—E2βŠ—β‹―βŠ—En)β†’Ξ“(F)\Gamma \left ( E_1 \otimes E_2 \otimes \cdots \otimes E_n \right ) \rightarrow \Gamma ( F )

In fact, there’s a natural isomorphism of sheaves of C∞C^\infty-modules Ξ“(βˆ’,E1)βŠ—CβˆžΞ“(βˆ’,E2)βŠ—Cβˆžβ‹―βŠ—CβˆžΞ“(βˆ’,En)β‰…Ξ“(βˆ’,E1βŠ—E2βŠ—β‹―βŠ—En)\Gamma ( - , E_1 ) \otimes_{C^\infty} \Gamma ( - , E_2 ) \otimes_{C^\infty} \cdots \otimes_{C^\infty} \Gamma ( - , E_n ) \cong \Gamma \left ( - , E_1 \otimes E_2 \otimes \cdots \otimes E_n \right ) 🀑

To recap: we’ve just shown that a tensor Ο„:Ξ“(E1)Γ—β‹―Ξ“(En)β†’Ξ“(F)\tau : \Gamma ( E_1 ) \times \cdots \Gamma ( E_n ) \rightarrow \Gamma ( F ) can be naturally identified with some Ο„βˆˆHomC∞(M)(Ξ“(E1βŠ—β‹―βŠ—En),Ξ“(F))\tau \in \text{Hom}_{C^\infty ( M )} \left ( \Gamma \left ( E_1 \otimes \cdots \otimes E_n \right ) , \Gamma ( F ) \right ). A natural question to ask ourselves at this point is: does Ο„\tau correspond to some Ο„βˆˆΞ“(Hom(E1βŠ—β‹―En,F))\tau \in \Gamma \left ( \text{Hom} \left ( E_1 \otimes \cdots E_n , F \right ) \right )? First of all, why does this make sense? Recall that given two vector spaces VV and WW, the set Hom(V,W)\text{Hom} ( V , W ) of linear transformations Vβ†’WV \rightarrow W is again a vector space. Hence we can consider the vector bundle Hom(E1βŠ—β‹―βŠ—En,F)β†’M\text{Hom} \left ( E_1 \otimes \cdots \otimes E_n , F \right ) \rightarrow M whose fibers are

Hom(E1βŠ—β‹―βŠ—En,F)|p=Hom(E1|pβŠ—β‹―βŠ—En|p,F|p)\text{Hom} \left ( E_1 \otimes \cdots \otimes E_n , F \right ) \text{|}_p = \text{Hom} \left ( E_1 \text{|}_p \otimes \cdots \otimes E_n \text{|}_p , F \text{|}_p \right )

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

i:Ξ“(Hom(E1βŠ—β‹―βŠ—En,F))β†’HomC∞(M)(Ξ“(E1βŠ—β‹―βŠ—En),Ξ“(F)),i : \Gamma \left ( \text{Hom} \left ( E_1 \otimes \cdots \otimes E_n , F \right ) \right ) \rightarrow \text{Hom}_{C^\infty ( M )} \left ( \Gamma \left ( E_1 \otimes \cdots \otimes E_n \right ) , \Gamma ( F ) \right ) ,

which takes Ξ·βˆˆΞ“(Hom(E1βŠ—β‹―βŠ—En,F))\eta \in \Gamma \left ( \text{Hom} \left ( E_1 \otimes \cdots \otimes E_n , F \right ) \right ) to iΞ·:Ξ“(E1βŠ—β‹―βŠ—En)β†’Ξ“(F)i \eta : \Gamma \left ( E_1 \otimes \cdots \otimes E_n \right ) \rightarrow \Gamma ( F ) with iΞ·(ΞΎ)p=Ξ·p(ΞΎp)i \eta {\left ( \xi \right )}_p = \eta_p \left ( \xi_p \right ) — notice this is precisely what we did to get from β€œa bilinear form in TpMT_p M for each p∈Mp \in M” to a Riemannian metric seen as a tensor. The meaning of β€œa transformation at each fiber pp that varies smoothly with pp” is now much clearer too: this is a smooth section of Hom(E1βŠ—β‹―βŠ—En,F)\text{Hom} \left ( E_1 \otimes \cdots \otimes E_n , F \right ). The inclusion ii is not surjective. This is because in general if Ο†:Ξ“(E1βŠ—β‹―βŠ—En)β†’Ξ“(F)\varphi : \Gamma \left ( E_1 \otimes \cdots \otimes E_n \right ) \rightarrow \Gamma ( F ) is a homomorphism the value of Ο†(ΞΎ1,…,ΞΎn)p\varphi(\xi^1, \ldots, \xi^n)_p may very well depend on ΞΎi\xi^i in their entirety, and not only on ΞΎpi\xi_p^i.

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

We claim, however, that the image of ii consists precisely of the multilinear functions E1Γ—β‹―Γ—Enβ†’FE_1 \times \cdots \times E_n \rightarrow F that are tensors — i.e. such that Ο„(ΞΎ1,…,ΞΎn)p\tau(\xi^1, \ldots, \xi^n)_p is determined by ΞΎpi\xi_p^i. Indeed, if we consider the map

s:T(E1Γ—β‹―Γ—En,F)β†’Ξ“(Hom⁑(E1βŠ—β‹―βŠ—En,F))s : \mathcal{T}(E_1 \times \cdots \times E_n, F) \to \Gamma(\operatorname{Hom}(E_1 \otimes \cdots \otimes E_n, F))

given by sΟ„p(v1,…,vn)=Ο„(ΞΎ1,…,ΞΎn)ps \tau_p (v_1, \ldots, v_n) = \tau(\xi^1, \ldots, \xi^n)_p, where T(E1Γ—β‹―Γ—En,F)βŠ‚Hom⁑C∞(M)(E1βŠ—β‹―βŠ—En,F)\mathcal{T}(E_1 \times \cdots \times E_n, F) \subset \operatorname{Hom}_{C^\infty(M)}(E_1 \otimes \cdots \otimes E_n, F) is the subspace of tensors and ΞΎiβˆˆΞ“(Ei)\xi^i \in \Gamma ( E_i ) are such that ΞΎpi=vi\xi_p^i = v_i, we can very quickly check that i=sβˆ’1i = s^{- 1}, establishing an isomorphism of C∞(M)C^\infty ( M )-modules

Ξ“(Hom⁑(E1βŠ—β‹―βŠ—En,F))β‰…T(E1Γ—β‹―Γ—En,F)\Gamma(\operatorname{Hom}(E_1 \otimes \cdots \otimes E_n, F)) \cong \mathcal{T}(E_1 \times \cdots \times E_n, F)

The definition of sΟ„p(v1,…,vn)s \tau_p (v_1, \ldots, v_n) does not depend on the choice of ΞΎi\xi^i precisely because the value of Ο„(ΞΎ1,…,ΞΎn)p\tau(\xi^1, \ldots, \xi^n)_p depends only on ΞΎpi=vi\xi_p^i = v_i! In conclusion, a tensor Ο„:E1Γ—β‹―Γ—Enβ†’F\tau : E_1 \times \cdots \times E_n \rightarrow F is called a tensor because it corresponds to a smooth section of Hom(E1βŠ—β‹―βŠ—En,F)\text{Hom} \left ( E_1 \otimes \cdots \otimes E_n , F \right ). 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(E1βŠ—β‹―βŠ—En,F)\text{Hom} \left ( E_1 \otimes \cdots \otimes E_n , F \right ) as E1βˆ—βŠ—β‹―βŠ—Enβˆ—βŠ—FE_1^* \otimes \cdots \otimes E_n^* \otimes F.

This is because given two vector spaces VV and WW, the space Hom(V,W)\text{Hom} ( V , W ) is canonically isomorphic to Vβˆ—βŠ—WV^* \otimes W. Taking V=E1|pβŠ—β‹―βŠ—En|pV = E_1 \text{|}_p \otimes \cdots \otimes E_n \text{|}_p and W=F|pW = F \text{|}_p, this translates to an isomorphism of vector bundles Hom(E1βŠ—β‹―βŠ—En,F)β†’E1βˆ—βŠ—β‹―βŠ—Enβˆ—βŠ—F\text{Hom} \left ( E_1 \otimes \cdots \otimes E_n , F \right ) \rightarrow E_1^* \otimes \cdots \otimes E_n^* \otimes F. In fact, usually the differential structure of Hom(E1βŠ—β‹―βŠ—En,F)\text{Hom} \left ( E_1 \otimes \cdots \otimes E_n , F \right ) is defined via the identification with E1βˆ—βŠ—β‹―βŠ—Enβˆ—βŠ—FE_1^* \otimes \cdots \otimes E_n^* \otimes F. This is the formalism generally adopted, which is to say, when a geometer says β€œa tensor” in a formal sense he most likely means β€œsome Ο„βˆˆΞ“(E1βˆ—βŠ—β‹―βŠ—Enβˆ—βŠ—F)\tau \in \Gamma \left ( E_1^* \otimes \cdots \otimes E_n^* \otimes F \right )”.

Also, if Fβ†’MF \rightarrow M is the trivial line bundle MΓ—RM \times \mathbb{R}, one usually refers to E1βˆ—βŠ—β‹―βŠ—Enβˆ—βŠ—FE_1^* \otimes \cdots \otimes E_n^* \otimes F by simply E1βˆ—βŠ—β‹―βŠ—Enβˆ—E_1^* \otimes \cdots \otimes E_n^*, because tensoring by MΓ—RM \times \mathbb{R} is the same as doing nothing. For instance, a Riemmanian metric is most often defined as a tensor gβˆˆΞ“(Tβˆ—MβŠ—Tβˆ—M)\text{g} \in \Gamma \left ( T^* M \otimes T^* M \right ) satisfying special conditions. That about wraps it up. I hope this helped someone πŸ˜›