# Geodesic preserving diffeomorphisms of constant curvature spaces

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.

## 1 Answer

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.

• 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. Aug 10, 2019 at 14:53