Project curves on a surface with no deformation

Thank you very much for your help. I know that it will be distort, thats why I spoke about “minimal deformation” :). I am still stuck. I dont think I made my point clear enough, so I’ll add some details. The only surface on which I got the result I was searching for (on the right), was taken from the rhino file in the solution of Pascal Golay on: Perfectly deforming curves around a sphere?. I dont know why mine (left one) is not working, what is the difference.