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.