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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 Majer'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...

Left above: $f(x,y)$. Right above: Level sets of $f$. Below: $\nabla f$.

And here (below) is a closeup of the function defined using squared distance $[d( p, C )]^2$, as per Willie Wong's suggestion: