Skip to content
DraconiticResearch
Search Sources API
194,164 studies31 sourcesUpdated 12 min ago
PreprintCC BY-NC-ND 4.0arXiv

Fixed geodesics of involutions on $3$-spheres

  • Bin Shen
  • Hui Zhang
Open at sourcePDFDOICite

Abstract

Let an isometric involution act on a smooth Riemannian $3$-sphere with sectional curvature at least one. We prove that every one-dimensional connected component of its fixed-point set has length at most 2π, and that the bound is sharp. The proof realizes the fixed circle as the link of a complete totally geodesic surface in a positively curved expanding Ricci soliton and applies Bishop--Gromov comparison to its area growth. The analytic input is an equivariant version of Deruelle's smoothing of positively curved cones. We use Gianniotis--Schulze's equivariant local deformation and explain a marked compactness argument for arbitrary finite isotropy. In particular, the end chart retains the specified group action, rather than only an unspecified conjugate.

Related

  1. A Fixed-Point Worpitzky Identity and a Positive Binomial Transform for Type B Involutions
    Jiang Zeng
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. Maximum Entropy Probability Distributions on Spheres with Fixed Mean Busemann Function and Holomorphic-Information-Geometric Model of Cognition
    Vladimir Jacimovic
    arXivPreprintCC BY 4.0Full textProbabilityarXiv
  3. Quivers with Involutions and Shifted Twisted Yangians via Coulomb Branches
    Yaolong Shen, Changjian Su, Rui Xiong
    Communications in Mathematical PhysicsPeer-reviewedLicence not statedFull text14 citedGeometry & topologyOpenAlex
  4. Some Coble surfaces admit only Bertini involutions
    Federico Pieroni
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Sylow $2$-Subgroups of Finite Groups with a Single Conjugacy Class of Involutions
    Tianze Li, Yu Zeng
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{shen2026fixed,
  title = {{Fixed geodesics of involutions on $3$-spheres}},
  author = {Bin Shen and Hui Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29796},
  eprint = {2610.29796},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29796},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Fixed geodesics of involutions on $3$-spheres
AU  - Bin Shen
AU  - Hui Zhang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29796
UR  - https://arxiv.org/abs/2610.29796
AB  - Let an isometric involution act on a smooth Riemannian $3$-sphere with sectional curvature at least one. We prove that every one-dimensional connected component of its fixed-point set has length at most 2π, and that the bound is sharp. The proof realizes the fixed circle as the link of a complete totally geodesic surface in a positively curved expanding Ricci soliton and applies Bishop--Gromov comparison to its area growth. The analytic input is an equivariant version of Deruelle's smoothing of positively curved cones. We use Gianniotis--Schulze's equivariant local deformation and explain a marked compactness argument for arbitrary finite isotropy. In particular, the end chart retains the specified group action, rather than only an unspecified conjugate.
ER  -