Beltrami problem in dimension two: local normal forms

Abstract

Two (pseudo-)Riemannian metrics are said to be geodesically equivalent if they share the same geodesics considered as unparametrized curves. In 1865, E. Beltrami posed the problem of describing all geodesically equivalent metric pairs locally. At generic points, this problem was solved by Dini in the Riemannian setting and later extended to the pseudo-Riemannian case by Bolsinov, Matveev, and Pucacco. In the present paper, we solve the Beltrami problem in dimension two by providing a complete local classification of geodesically equivalent pseudo-Riemannian metrics in a neighbourhood of a singular point. This yields a full solution of the problem in dimension two.