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Sophus Lie's problem on two-dimensional metrics with projective symmetries: completing the local classification

  • Gianni Manno
  • Vladimir S. Matveev
  • Filippo Salis
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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.

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Cite

BibTeXRIS
BibTeX
@misc{manno2026sophus,
  title = {{Sophus Lie's problem on two-dimensional metrics with projective symmetries: completing the local classification}},
  author = {Gianni Manno and Vladimir S. Matveev and Filippo Salis},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05280},
  eprint = {2609.05280},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05280},
}
RIS
TY  - UNPB
TI  - Sophus Lie's problem on two-dimensional metrics with projective symmetries: completing the local classification
AU  - Gianni Manno
AU  - Vladimir S. Matveev
AU  - Filippo Salis
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05280
UR  - https://arxiv.org/abs/2609.05280
AB  - 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.
ER  -