What is the Tensor Analog of the Poincare Lemma?

[Ancient_Nomad]

Hi everyone,

I know that there is a result that corresponding to a closed p-form α, I can find a p-1 form β,
such that dβ = α.

I wanted to ask, what the tensorial analog of this would be. I mean would it be right to say that on a manifold with a lorentzian metric,

If I have a vector field A, such that

\nabla_{\mu} A^{\mu} = 0

then there exists an anti-symmetric tensor B such that,

\nabla_{\nu} B^{\mu \nu} = A^{\mu}

where \nabla is the covariant derivative comaptible with the metric
(So that covariant derivative of the metric is zero)

Also, if possible, please tell me where I can find the proof of the correct statement.

Thanks in advance.

 

[holy_toaster]

I think what you are looking for is a Poincare Lemma for the co-differential, if I get this right. So the problem is not so much about using vector fields or forms, when you have a metric available (as you say) to switch between them.

Essentially the divergence \mathrm{div}(A) of a vector field A is the co-differential \delta(A^{\flat}) of its corresponding one-form A^{\flat}. As the co-differential is also nil-potent as is the exterior derivative, there is of course also a Poincare Lemma for the co-differential. Something like: Every co-closed form is co-exact.

But note that these statement generally only hold locally on manifolds with non-vanishing first Betti number.

 

[Ben Niehoff]

On any Riemannian manifold one has the Hodge decomposition, which states that any n-form \varphi_n can be written

\varphi_n = d \alpha_{n-1} + \delta \beta_{n+1} + \gamma_n
for some \alpha_{n-1}, \beta_{n+1}, \gamma_n (the subscript indicates the degree of the form), where \delta is the co-differential, and \gamma_n is harmonic (i.e. d \gamma_n = \delta \gamma_n = 0).

If we assume that \varphi_n is co-closed, then

\delta \varphi_n = 0 = \delta d \alpha_{n-1} + 0 + 0
hence we must have \alpha_{n-1} = 0. If the n-th Betti number is zero, then we must also have \gamma_n = 0, and in this case \varphi_n can be written

\varphi_n = \delta \beta_{n+1}
This statement is always true locally.