# Geodesic-like curves stemming from the heat kernel on a manifold

Consider a smooth $n$-dimensional Riemannian manifold $M$ with sufficiently nice geometric and topological properties such that there exist a unique heat kernel $(t,x,y) \mapsto p(t,x,y)$ on it ($t>0$). Consider the equality

$$p(t,x,y) = \frac 1 {\sqrt {4 \pi t}^n} \Bbb e ^{- \frac 1 4 \int \limits _0 ^t \langle \dot c, \dot c \rangle \ \Bbb d s } .$$

If $M = \Bbb R^n$, the curve $c$ satisfying the above equality is precisely the (unique) geodesic connecting $x$ and $y$.

My question is: what can be said about $c$ on abstract manifolds $M \ne \Bbb R ^n$? Do these "fake geodesics" have a name, have they been studied? If so, where could I find more information about them?

Clearly they cannot be geodesics, in general, since the equality

$$\int \limits _0 ^t \langle \dot c, \dot c \rangle \Bbb d s = \frac {d(x,y)^2} t$$

for a geodesic $c: [0,t] \to M$ with $c(0) = x$ and $c(0) = y$

would imply that

$$p(t,x,y) = \frac 1 {\sqrt {4 \pi t}^n} \Bbb e ^{- \frac {d(x,y)^2} {4t} } ,$$

which in general is not true.

I don't think they have a name. Also for say $2$-manifolds, they are not unique in negative curvature, and don't exist in positive curvature.
Indeed, in positive curvature, the heat kernel will be larger than for the plane, but maximizing the expression you suggest w.r.t to $c$ will give the formula you wrote for the planar heat kernel.