Summary: Let \(M\) be a compact, connected Riemannian manifold, and let \(p_t(x,y)\) denote the fundamental solution to Cauchy initial value problem for the heat equation \({\partial u\over\partial t}={1\over 2} \Delta u\), where \(\Delta\) is the Levi-Civita Laplacian. The purpose of this note is to describe the behavior of the Hessian of \(\log p_T(\cdot,y)\) for small \(T>0\). Emphasis is given to the difference between what happens outside, where the behavior is like \({1\over T}\), as opposed to at the cut locus, where it is like \({1\over T^2}\).