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.