A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields

Abstract

We give a complete list of normal forms for the 2-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie.

Complete Einstein Metrics are Geodesically Rigid

“…The tensor R ik,j −R ij,k is called projective Yano tensor, and plays important role in the theory of geodesically equivalent metrics; in particular, it is projectively invariant in dimension 2 [23,9], and is an essential part of the so-called tractor approach for the investigation of geodesically equivalent metrics [13].…”

Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics

“…The reason for this is that in dimension 2 the Lie problem was solved in [BMM,Ma7], where a complete list of local metrics admitting projective vector fields was constructed. The list is pretty simple (explicit formulas involving only elementary functions), and it should not be very complicated to understand which metrics from this list can be prolonged up to a complete metric.…”

“…In the Riemannian case, equation (18) was obtained by other methods and played an important role in the proof of the projective Lichnerowicz Conjecture [Ma5], and in the local description of projective vector fields [Fu,So,BMM,Ma7]. This equation is relatively simple and can be solved in the Riemannian case explicitly.…”

Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

“…Theorem 3 had found a recent important application in the solution of two problems explicitly stated by Sophus Lie in [25] due to [12,35].…”