Kolmogorov-Fokker-Planck operators in dimension two: heat kernel and curvature

We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov-Fokker-Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants of the optimal control problem associated with the diffusion. This gives a first example of geometric interpretation of the small-time heat kernel asymptotics of non-homogeneous H\"ormander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order part does not satisfy the H\"ormander condition.