Errata on Tensors

February this year I posted some comments on how tensors show up in geometry. However, having enrolled in Claudio Gorodski's course in Riemannian Geometry at IME-USP, I now know a thing or two about tensors I didn’t knew at the time. I’m proud to say that most of what I said stands up to the test of time, but I would still like to make some corrections.

If you’re interested in Riemannian geometry, I strongly recommend Claudio’s course notes.

First and foremost, in the previous post I explicitly stated that there are vector bundles Ei→ME_i \rightarrow M and F→MF \rightarrow M such that the map

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 )

that takes a tensor TβˆˆΞ“(Hom(E1βŠ—β‹―βŠ—En,F))T \in \Gamma \left ( \text{Hom} \left ( E_1 \otimes \cdots \otimes E_n , F \right ) \right ) to the homomorphism iT:Ξ“(E1βŠ—β‹―βŠ—En)β†’Ξ“(F)i T : \Gamma \left ( E_1 \otimes \cdots \otimes E_n \right ) \rightarrow \Gamma ( F ) given by (iT)(ΞΎ1,…,ΞΎn)p=Tp(ΞΎp1,…,ΞΎpn)(i T)(\xi^1, \ldots, \xi^n)_p = T_p(\xi_p^1, \ldots, \xi_p^n) is not surjective. This is not the case: every C∞(M)C^\infty ( M )-multilinear map T:Ξ“(E1)Γ—β‹―Γ—Ξ“(En)β†’Ξ“(F)T : \Gamma ( E_1 ) \times \cdots \times \Gamma ( E_n ) \rightarrow \Gamma ( F ) is the image of some tensor, which is to say, the value of T(ΞΎ1,…,ΞΎn)pT(\xi^1, \ldots, \xi^n)_p depends only on the values of ΞΎpi\xi_p^i.

This result is known as the tensor field characterization lemma, and a proof can be found in here. This particular proof is specific to the case where each Ei=TME_i = T M and F=MΓ—RF = M \times \mathbb{R} — which is to say, the proof is specific to the case where T:X(M)Γ—β‹―Γ—X(M)β†’C∞(M)T : \mathfrak{X} ( M ) \times \cdots \times \mathfrak{X} ( M ) \rightarrow C^\infty ( M ) — but I believe the same argument should work for sections of arbitrary smooth vector bundles. The proof, however, is heavily reliant on the existence of smooth partitions of unity, which hints at the fact that perhaps the lemma does not hold for holomorphic vector fields over complex manifolds — i.e. perhaps there some complex manifold XX with holomorphic vector fields Eiβ†’XE_i \rightarrow X and an OX\mathcal{O}_X-multilinear map T:Ξ“(E1)Γ—β‹―Γ—Ξ“(En)β†’Ξ“(F)T : \Gamma ( E_1 ) \times \cdots \times \Gamma ( E_n ) \rightarrow \Gamma ( F ) which is not the image of a holomorphic tensor.

The fact that this proof is restricted to the particular case where Ei=TME_i = T M and F=MΓ—RF = M \times \mathbb{R} is not a coincidence: in most cases we are interested in tensors with the signature X(M)Γ—β‹―Γ—X(M)Γ—Ξ©1(M)Γ—β‹―Γ—Ξ©1(M)β†’C∞(M)\mathfrak{X} ( M ) \times \cdots \times \mathfrak{X} ( M ) \times \Omega^1 ( M ) \times \cdots \times \Omega^1 ( M ) \rightarrow C^\infty ( M ). The space of tensors which take ss fields and rr 1-forms is usually referred to as T(r,s)M\mathcal{T}^{(r, s)} M. In particular, if MM is a Riemannian manifold with metric g\mathrm{g}, the canonical isomorphism TpMβ†’Tpβˆ—MT_p M \rightarrow T_p^* M given by the metric gp{\mathrm{g}}_p induces an isomorphism of C∞(M)C^\infty ( M )-modules X(M)β†’Ξ©1(M)\mathfrak{X} ( M ) \rightarrow \Omega^1 ( M ).

Another way to put it is to say that g\mathrm{g} induces an isomorphism T(0,1)Mβ†’T(1,0)M\mathcal{T}^{(0, 1)} M \to \mathcal{T}^{(1, 0)} M, which in turn induces the so called musical isomorphisms β™―:T(s,r+1)Mβ†’T(s+1,r)M\sharp : \mathcal{T}^{(s, r + 1)} M \to \mathcal{T}^{(s + 1, r)} M and β™­:T(s+1,r)Mβ†’T(s,r+1)M\flat : \mathcal{T}^{(s + 1, r)} M \to \mathcal{T}^{(s, r + 1)} M. In particular, T(r,s)M\mathcal{T}^{ (r, s) } M is canonically isomorphic to T(0,r+s)M\mathcal{T}^{ (0, r + s) } M via β™―s\sharp^s, which is the space of tensors with signature X(M)Γ—β‹―Γ—X(M)β†’C∞(M)\mathfrak{X} ( M ) \times \cdots \times \mathfrak{X} ( M ) \rightarrow C^\infty ( M ) — so that the proof of the lemma for this particular case is sufficient. That about wraps it up. Hope this helped someone πŸ˜›