I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies $$p_t(x, y) = \frac{1}{(4\pi t)^{n/2}}e^{-\frac{\text{dist }(x, y)^2}{4t}} + O(e^{-\frac{1}{\sqrt{t}}}).$$

Is this correct? Where can I find a reference for this fact? Thanks!