Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 4 min ago
PreprintarXiv licencearXiv

Rotationally Symmetric Zoll Metrics with a Cubic Integral

  • Holger R. Dullin
  • Vladimir S. Matveev
  • Gleb P. Palshin
  • Serena Scapucci
Open at sourcePDFDOICite

Abstract

We introduce a new method for constructing Zoll metrics that admit first integrals polynomial in the momenta. The method uses Funk's form of the metric together with an algebraic relation between the first integrals. This reduces the problem to an algebraic system from which all unknown functions can be determined explicitly. In the case of a non-trivial cubic integral, we present an explicit two-parameter family of metrics that covers all such rotationally symmetric Zoll metrics on the $2$-sphere: every such metric is isometric to a member of this family. We compare these metrics with the earlier results of Valent, Duval, and Shevchishin.

Related

  1. On Rotationally Symmetric Norms
    Yossi Lonke
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  2. Seifert type magnetohydrostatic equilibria in rotationally symmetric toroidal domains
    Daniel Peralta-Salas, Radu Slobodeanu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsMathematical physicsarXiv
  3. Almgren frequency and Agmon convexity on rotationally symmetric Cartan-Hadamard manifolds
    Luca F. Di Cerbo
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  4. On convex bodies with rotationally symmetric planar projections
    Sergii Myroshnychenko, Dmitry Ryabogin, Christos Saroglou
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryAnalysis & PDEsarXiv
  5. Existence of rotationally symmetric embedded f-minimal tori
    Peng Peng
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{dullin2026rotationally,
  title = {{Rotationally Symmetric Zoll Metrics with a Cubic Integral}},
  author = {Holger R. Dullin and Vladimir S. Matveev and Gleb P. Palshin and Serena Scapucci},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.31481},
  eprint = {2609.31481},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31481},
}
RIS
TY  - UNPB
TI  - Rotationally Symmetric Zoll Metrics with a Cubic Integral
AU  - Holger R. Dullin
AU  - Vladimir S. Matveev
AU  - Gleb P. Palshin
AU  - Serena Scapucci
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.31481
UR  - https://arxiv.org/abs/2609.31481
AB  - We introduce a new method for constructing Zoll metrics that admit first integrals polynomial in the momenta. The method uses Funk's form of the metric together with an algebraic relation between the first integrals. This reduces the problem to an algebraic system from which all unknown functions can be determined explicitly. In the case of a non-trivial cubic integral, we present an explicit two-parameter family of metrics that covers all such rotationally symmetric Zoll metrics on the $2$-sphere: every such metric is isometric to a member of this family. We compare these metrics with the earlier results of Valent, Duval, and Shevchishin.
ER  -