We prove that if a connected Lie group acts on a connected complete Riemannian surface of nonconstant curvature by diffeomorphisms that take (unparameterised) geodesics to geodesics, then it acts by isometries.
Cite this article
Vladimir S. Matveev, Lichnerowicz-Obata conjecture in dimension two. Comment. Math. Helv. 80 (2005), no. 3, pp. 541–570DOI 10.4171/CMH/25