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Singularities of Separation Coordinates

  • Dinmukhammed Akpan
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Abstract

We study singularities of separation coordinates for geodesic flows of real-analytic two-dimensional (pseudo-)Riemannian metrics. We obtain a complete local classification of these singularities using the Nijenhuis operator associated with a quadratic first integral. The singular set is locally a smooth curve, an isolated point, or two transversely intersecting curves. We also show that orthogonal separation near one point of an analytic disk extends to a quadratic first integral on the entire disk and provide explicit separation coordinates near singularities.

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Cite

BibTeXRIS
BibTeX
@misc{akpan2026singularities,
  title = {{Singularities of Separation Coordinates}},
  author = {Dinmukhammed Akpan},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02939},
  eprint = {2611.02939},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02939},
}
RIS
TY  - UNPB
TI  - Singularities of Separation Coordinates
AU  - Dinmukhammed Akpan
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.02939
UR  - https://arxiv.org/abs/2611.02939
AB  - We study singularities of separation coordinates for geodesic flows of real-analytic two-dimensional (pseudo-)Riemannian metrics. We obtain a complete local classification of these singularities using the Nijenhuis operator associated with a quadratic first integral. The singular set is locally a smooth curve, an isolated point, or two transversely intersecting curves. We also show that orthogonal separation near one point of an analytic disk extends to a quadratic first integral on the entire disk and provide explicit separation coordinates near singularities.
ER  -