For vector fields in $R^3$ one knows that there exists a unique decomposition of vector fields in to solenoidal (divergence free) and potential parts. A generalization of this Theorem for symmetric tensors of arbitrary order is known for compact Riemannian manifolds with boundary , e.g [Theorem 3.3.2], Sharafutdinov, *Integral Geometry of Tensor Fields*.

I am curious if such a Theorem is known for any class of Lorentzian Manifolds $M$ too (in particular for Space-time)? i.e. Assume $f$ is a vector field (or more generally a symmetric tensor of order $m$), then does there exist a divergence free vector field (or a symmetric tensor of order $m$) (solenoidal) $f^s$ and a function (or a $m-1$ form) $v$ such that $f=f^s+dv$ , $v|_{\partial M} = 0 $ and where $d$ represents the operator which takes symmetrized co variant derivatives.

**As a first attempt at proof**, following Sharafutdinov's approach, we will say that If such a decomposition were to exist then we should be able to find a unique solution to the PDE system $\delta d (v) = \delta f$, $v|_{\partial M} = 0 $. Here $\delta$ represents the divergence operator. Now, if we calculate the symbol for $\delta d$, we can show that at least for the case when $v$ is a function, the operator $\delta d$ is in fact strictly hyperbolic. For higher order tensors, $\delta d$ fails to be strictly hyperbolic. So the question really reduces to the following : When do we know that a solution for a hyperbolic system $Lv=f$, $v|_{\partial M} = 0 $ exists and is unique?

Also, in the Space-time setting, it looks more natural to work in a non-compact manifold (at least which go in future time directions forever e.g. like a $[0,\infty] \times N$ manifold where $N$ is a compact Riemannian Manifold. Note that Sharafutdinov's theorem is really only for compact Riemannian manifolds, then does it make sense for us to ask a similar question for a non-compact Lorentzian manifolds? Any ideas/comments are welcome as I am feeling out of my depth here!