On the Heat Diffusion for Generic Riemannian and Sub-Riemannian Structures

In this article, we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: Generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point), and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As a by-product, we show that, for generic low-dimensional Riemannian manifolds, the only singularities of the exponential map, as a Lagragian map, that can arise along a minimizing geodesic are A3 and A5 (in the classification of Arnol'd's school). We show that in the nongeneric case, a cornucopia of asymptotics can occur, even for Riemannian surfaces.