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Let $X$ be either Euclidean space $\mathbb{R}^n$, the sphere $\mathbb{S}^n$, or hyperbolic space $\mathbb{H}^n$.

I would like to have a classification of all diffeomorphisms $X\to X$ which map every geodesic line to a geodesic line.

In the first two cases, the group of all such transformations is strictly larger than the group of isometries, but for the hyperbolic space I am not sure.

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For $\mathbb{R}^n$: the fundamental theorem of projective geometry (proof: https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf) says that the bijections of $\mathbb{R}^n$ taking lines to lines are the affine maps $x\mapsto Ax+b$ for an invertible matrix $A$ and a constant vector $b$.

For $S^n$: a theorem by the same name shows that the bijections of projective space taking projective lines to projective lines is a projective transformation. One easily uses this to prove the result for the sphere that such a bijection is a linear transformation defined up to positive rescaling, acting on the sphere as the space of rays in a real vector space.

For $\mathbb{H}^n$: Kobayashi defined a Kobayashi pseudometric for projective connections, which determines the usual metric when applied to hyperbolic space, so the unparameterized geodesics determine the metric, and so the diffeomorphisms preserving them are isometries:

S. Kobayashi, Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. Tokyo IA 24 (1977), 129--135.

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    $\begingroup$ You could also use Hilbert's construction of metrics from cross ratios to prove the result for hyperbolic space, if I remember correctly, and that might get around the need to assume diffeomorphism instead of just bijection. $\endgroup$ – Ben McKay Aug 10 '19 at 14:53

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