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
EiββM
and
FβM
such that the map
that takes a tensor
TβΞ(Hom(E1βββ―βEnβ,F))
to the homomorphism
iT:Ξ(E1βββ―βEnβ)βΞ(F)
given by
(iT)(ΞΎ1,β¦,ΞΎn)pβ=Tpβ(ΞΎp1β,β¦,ΞΎpnβ)
is not surjective. This is not the case: every
Cβ(M)-multilinear map
T:Ξ(E1β)Γβ―ΓΞ(Enβ)βΞ(F)
is the image of some tensor, which is to say, the value of
T(ΞΎ1,β¦,ΞΎn)pβ
depends only on the values of
ΞΎpiβ.
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
Eiβ=TM
and
F=MΓR — which is to say, the proof is specific to the case
where
T:X(M)Γβ―ΓX(M)βCβ(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
X
with holomorphic vector fields
EiββX
and an
OXβ-multilinear map
T:Ξ(E1β)Γβ―ΓΞ(Enβ)βΞ(F)
which is not the image of a holomorphic tensor.
Another way to put it is to say that
g
induces an isomorphism
T(0,1)MβT(1,0)M, which in turn induces the so called musical isomorphismsβ―:T(s,r+1)MβT(s+1,r)M
and
β:T(s+1,r)MβT(s,r+1)M. In particular,
T(r,s)M
is canonically isomorphic to
T(0,r+s)M
via
β―s, which is the space of tensors with signature
X(M)Γβ―ΓX(M)βCβ(M) — so that the proof of the lemma for this particular
case is sufficient. That about wraps it up. Hope this helped someone π