I am looking for a differentiable map $f: S^n\rightarrow S^n$, which conserves the geodetic lines of the standard metric on $S^n$, but is no isometry.
The geodetic lines on $S^n$ should be the great circles, I think.
Unfortunately, I was only able to think of examples, which are not well-defined...
I would be grateful for any help.
$S^n$ can be identified with $\mathbb{R}^{n+1}/\mathbb{R}_+$ (where $\mathbb{R}_+$ acts on $\mathbb{R}^{n+1}$ by multiplication). Geodesics come from $2$-dimensional vector subspaces of $\mathbb{R}^{n+1}$. The group $GL(n+1,\mathbb{R})$ acts on $\mathbb{R}^{n+1}/\mathbb{R}_+$ and maps geodesics to geodesics. If you take a matrix $A\in GL(n+1,\mathbb{R})$ which is not a constant multiple of an orthogonal matrix, then its action does not preserve the metric.