A note on the structure of prescribed gradient–like domains of non–integrable vector fields

Abstract

Given a geometric structure on with (e.g., Euclidean, symplectic, Minkowski, and pseudo‐Euclidean), we analyze the set of points inside the domain of definition of an arbitrary given vector field, where the value of the vector field equals the value of the left/right gradient–like vector field of some fixed potential function, although a natural non‐integrability condition holds at each such a point. More precisely, we prove that if not empty, this is a Borel set which admits a cover consisting of at most regularly embedded m−dimensional submanifolds of , and consequently, its Hausdorff dimension is less or equal to m. Particular examples of gradient–like vector fields include the class of gradient vector fields with respect to Euclidean or pseudo‐Euclidean inner products, and the class of Hamiltonian vector fields associated to symplectic structures on . The main result of this article provides a geometric version of the similar result for the classical gradient vector field.