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...
<br />
![Function, Contours][1]
<br />
_Left above_: $f(x,y)$. _Right above_: Level sets of $f$.
_Below_: $\nabla f$.
<br />
![Gradient][2]
<br />
And here (_below_) is a closeup of the function defined using squared distance
$[d( p, C )]^2$, as per Willie Wong's suggestion:
<br />![WW][3]


  [1]: https://i.sstatic.net/mwvrR.jpg
  [2]: https://i.sstatic.net/6Sp7s.jpg
  [3]: https://i.sstatic.net/A5gpI.jpg