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Here is a function $f(x,y)$ which is 0 inside the square $C=[\pm1,\pm1]$, and outside that square has value equal to the Euclidean distance $d( p, C )$ from $p=(x,y)$ to the boundary of $C$. [I am trying to follow Pietro's suggestion, as far as I understand it.] It is not a surface of revolution (but it is centrally symmetric). Are its gradient descent paths geodesics? I think so...
Function, Contours http://cs.smith.edu/%7Eorourke/MathOverflow/GradientSquare.jpg
Left above: $f(x,y)$. Right above: Level sets of $f$. Below: $\nabla f$.
Gradient http://cs.smith.edu/%7Eorourke/MathOverflow/VectorFieldSquare.jpg