Consider the closed surface $M$ embedded in $\Bbb R^3$ with $g:=ds^2=dx^2+dy^2+dz^2$ and with $M:=\log^2 x+ \log^2 y +\log^2 z=1.$
Then restrict the metric to $M$. Here is a 3D plot of $M$ embedded in $\Bbb R^3:$
For the unit sphere $S^2$ the closed geodesics all have the same length (traversing the loop once).
Do all the closed geodesics on $M$ have the same length?. A diagram/picture that illustrates the geodesic on $M$, perhaps supplemented with some mathematics would be great. I'm looking for the big picture here.
I've investigated the case for $S^2$ with the restricted metric $ ds^2=\frac{dr^2}{1-r^2}+r^2 d\theta^2. $ This leads to great circles as geodesics. Traversing these loops once suggests they all have the same length and that there are infinitely many of them.
I sense that something similar should be happening for $M$. By a result of Birkhoff, there are infinitely many closed geodesics on $M$. However, I don't think the geodesics on $M$ will be as simple as was for the case on $S^2.$ I think the geodesics will have lengths bounded by certain constants.
I've also investigated the case where the ambient space $N:=\Bbb R^3_{\gt 0}$ is endowed with the metric $g':=ds^2= \frac{dx^2}{x^2}+\frac{dy^2}{y^2}+\frac{dz^2}{z^2}$ where I set up an isometry between $(\Bbb R^3,g)$ and $(N,g')$ given by $T:N \to \Bbb R^3$ with $T(x,y,z)= (\log x,\log y,\log z) .$ If one works like this, then you can just use the isometry to transport geodesics back and forth from $\Bbb R^3$ to $N$ which is not as interesting.
Of course this doesn't address the problem at hand but it does provide some valuable context.
