Sum of squares manifolds: The expressibility of the Laplace-Beltrami operator on pseudo-Riemannian manifolds as a sum of squares of vector fields

In this paper, we investigate under what circumstances the Laplace-Beltrami operator on a pseudo-Riemannian manifold can be written as a sum of squares of vector fields, as is naturally the case in Euclidean space.

We show that such an expression exists globally on one-dimensional manifolds and can be found at least locally on any analytic pseudo-Riemannian manifold of dimension greater than two. For two-dimensional manifolds this is possible if and only if the manifold is flat.

These results are achieved by formulating the problem as an exterior differential system and applying the Cartan-Kähler theorem to it.