Finding a metric in $ \Bbb R^2 $ depending on $s$ such that $x^s+y^s=1$ is a geodesic for all $s$ wrt. the metric

Question

Looking for a Riemannian metric or Pseudo-Riemannian metric in $ \Bbb R^2 $ depending on $s$ such that $x^s + y^s = 1$ is a geodesic for all $s$ wrt. the metric. $x,y\in(0,1), s\in \Bbb R(0, \infty). $

Edit: The points at the axes are problematic but they should be fine given the open interval for $x$ and $y$.

Thanks.

Answer 1

Maybe I'm missing something, but if, per the comment, there is no real regularity requirement after all, then I think the following works. Write down a change of coordinates $f$ that takes this curve to the first-quadrant arc of the unit circle, e.g., $f:(x,y)\mapsto (x^{s/2},y^{s/2})$. Write down any metric such that the unit circle is a geodesic, for example $ds^2=dr^2+d\theta^2$. Invert $f$ and transform the metric according to $f^{-1}$.