Let $f:M \to N$ a diffeomorphism between riemannian manifolds of the same dimension. What are sufficient conditions for $f$ to map geodesics to geodesics? Of course, if $f$ is an isometry this occurs, but I am looking for weaker conditions.
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Do you know of an example of such an $f$ on a compact manifold $M$ which is NOT an isometry? – Jason DeVito - on hiatus May 14 '16 at 23:30
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1If you scale $\mathbb R^n$ and restrict to the sphere you get such a map – Spotty May 16 '16 at 09:40
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In general, there are manifolds with self-diffeomorphisms which map geodesics to geodesics, but are not isometries. For instance, any affine map on $\mathbb{R}^n$ (i.e multiplication by a constant matrix) does this.
See also here.
Asaf Shachar
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Another example is the Klein model of the hyperbolic space: Hyperbolic geodesics map to euclidean geodesics. – Moishe Kohan May 16 '16 at 16:41