Let $(M,g,V)$ be a Riemannian manifold with potential $V\in\mathcal{C}^{\infty}(M)$. Let $\gamma : I\to M$ be a smooth curve, $I\subseteq \mathbb{R}$ an open interval. We say that $\gamma$ is the trajectory of a constrained unit-mass point-particle (or, simply, a *constrained trajectory*) if

$\nabla_{\dot{\gamma}}\dot{\gamma}=\langle X, \nu \rangle \cdot \nu $

for every $t\in I$, where $\nabla$ is the Levi-Civita connection, $X:=\mathrm{grad}_gV$ is the "force of gravity" and $\nu:=\dot{\gamma}/||\dot{\gamma}||$ is the tangent direction (whenever $\dot{\gamma}\neq 0$). I think this definition reduces to the "standard" one when $M$ is flat $\mathbb{R}^3$ and $V=-x_3$ (if it doesn't, please correct it so that it fits the physical viewpoint).

Let's say a constrained trajectory $\gamma$ is a *brachistochrone* between points $a, b\in M$, with $V(a)>V(b)$, if

1) $\gamma (0)=a$, $\dot{\gamma}(0)=0$, there is $T>0$ with $\gamma(T)=b$, and

2) for every other constrained trajectory $\eta$, with $\eta(0)=a$ and $\dot{\eta}(0)=0$ with $\eta(T^{*})=b$ for $T^{*}>0$, we have $T\leq T^{*}$.

Has this concept already been studied? Does it have any interesting applications?

Can anything interesting be said if $M$ is, say, the round $2$-sphere in $\mathbb{R}^3$ and $V$ is $-x_3$? Or what in the case $M$ is homogeneus (such as the hyperbolic plane) ?

*Note:* this question was just out of curiosity, I'm not doing research on this.