  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.
