In $\mathbb{R}^n$, if $\gamma$ is a line segment between $x_0 = \gamma (0)$ and $x = \gamma (t)$, one has the following formula:

$$\frac {\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, \dot{\gamma}> \mathbb{d}s}}{\sqrt {4 \pi t}^n} = h(t,x,x_0) ,$$

where $h$ is the kernel of the heat equation centered in $x_0$ at time $0$.

Now, if $(M,g)$ is a Riemann manifold (with the appropriate conditions for the heat kernel to exist), is there any curve $\gamma$ that could play a similar role to the line segment in the above formula (possibly with a different denominator)?

The only formula that I know of is

$$\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, \dot{\gamma}> \mathbb{d}s} = \mathbb{e}^ {- \frac{d(x,x_0)^2}{4 t}} ,$$

where $d$ is the distance associated to $g$ and $\gamma$ is a *minimizing geodesic*. This is nice, but quite far from the heat kernel.