What is the group of symmetries of $\mathbb{R^n}$ with the flat projective structure?

Question

Consider $X = (\mathbb{R^n},c)$, where $c$ is the equivalence class of all torsion free affine connections having straight lines as unparameterized geodesics. What is the group of symmetries of $X$? This ought to be known. Thank you.

Answer 1

Long answer taking a complicated but rewarding detour: This is example that fits into the framework of Cartan geometries. Category of manifolds with an equivalence class of torsion free affinne connections that have the same unparametrized geodesics is equivalent to (properly normalized) Cartan geometry modeled on a homogeneous space $G / P$ where $G = SL(n+1, \mathbb{R})$ and $P$ is the (parabolic) subgroup of $G$ that stabilizes a line in $\mathbb{R}^{n+1}$. By the fundamental theorem of Cartan geometries, if the Cartan curvature is zero then the structure is locally isomorphic to open subset of the homogeneous space $G/P$ (and the automorphism group is maximal up to "discrete part"). If you calculate the Cartan curvature in your case you indeed get zero. The proof of the fundamental theorem of Cartan geometries even gives you the isomorphism kind of explicitly via development map. Since your manifold is simply connected, the development map should provide global isomorphism onto an open subset of $G/P \simeq \mathbb{R}P^n.$ It remains to check which subgroup of $G$ fixes this subset.