Sophus Lie's problem on two-dimensional metrics with projective symmetries: completing the local classification
Year: 2026
Type: preprint
Abstract: We complete the local classification, up to isometry, of $2$-dimensional pseudo-Riemannian metrics (i.e. both Riemannian and Lorentzian), admitting a projective symmetry algebra of dimension at least two. The new contribution is the treatment of the non-regular case: we obtain a complete list of mutually non-isometric normal forms in neighbourhoods of points where the action of the projective Lie symmetry algebra fails to be regular, including all singular behaviours that can occur. Together with the known classification in the regular case, this completes the solution of the problem posed by Sophus Lie in 1882.
Source: arXiv (Cornell University)
Institutions: Politecnico di Torino, Friedrich Schiller University Jena
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211904339