Let $f(x,y)$ define a surface $S$
in $\mathbb{R}^3$ with a unique local minimum at $b \in S$.
Suppose gradient descent from any start point $a \in S$
follows a geodesic on $S$ from $a$ to $b$.
(**Q1**.)
What is the class of functions/surfaces
whose gradient-descent paths are geodesics?

Certainly if $S$ is a surface of revolution about a $z$-vertical line through $b$, its "meridians" are geodesics, and these would be the paths followed by gradient descent down to $b$. So the class of surfaces includes surfaces of revolution. But surely it is wider than that?

(**Q2**.)
One could ask the same question about paths followed by
Newton's method, which in general are different from gradient-descent
paths, as indicated in this Wikipedia image:

*Gradient descent: green.
Newton's method: red.*

(**Q3**.) These questions make sense in arbitrary dimensions,
although my primary interest is for surfaces in $\mathbb{R}^3$.

Any ideas on how to formulate my question as constraints on $f(\;)$, or pointers to relevant literature, would be appreciated. Thanks!

Q4.) Let $h:{\mathbb R}\rightarrow{\mathbb R}$ be smooth and increasing. Let $f$ be a function as in (Q1). Is $h\circ f$ such a function too ? – Denis Serre Oct 18 '10 at 13:55