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 EiM and FM such that the map

i:Γ(Hom(E1En,F))HomC(M)(Γ(E1En),Γ(F))

that takes a tensor TΓ(Hom(E1En,F)) to the homomorphism iT:Γ(E1En)Γ(F) given by (iT)(ξ1,,ξn)p=Tp(ξp1,,ξpn) is not surjective. This is not the case: every C(M)-multilinear map T:Γ(E1)××Γ(En)Γ(F) is the image of some tensor, which is to say, the value of T(ξ1,,ξn)p depends only on the values of ξpi.

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=TM and F=M× — which is to say, the proof is specific to the case where T:X(M)××X(M)C(M) — but I believe the same argument should work for arbitrary smooth vector fields. 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 X with holomorphic vector fields EiX and an OX-multilinear map T:Γ(E1)××Γ(En)Γ(F) which is not the image of a holomorphic tensor.

The fact that this proof is restricted to the particular case where Ei=TM and F=M× is not a coincidence: in most cases we are interested in tensors with the signature X(M)××X(M)×Γ(T*M)××Γ(T*M)C(M). The space of tensors which take s fields and r cofields is usually referred to as T(r,s)M. In particular, if M is a Riemannian manifold with metric g, the canonical isomorphism TpMTp*M given by the metric gp induces an isomorphism of C(M)-modules X(M)Γ(T*M).

Another way to put it is to say that g induces an isomorphism T(0,1)MT(1,0)M, which in turn induces the so called musical isomorphisms :T(s,r+1)MT(s+1,r)M and :T(s+1,r)MT(s,r+1)M. In particular, T(r,s)M is canonically isomorphic to T(0,r+s)M via s, which is the space of tensors with signature X(M)××X(M)C(M) — so that the proof of the lemma for this particular case is sufficient. That about wraps it up. Hope this helped someone 😛