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Geodesic Divergence on Riemannian Planes with Bounded Geometry

  • R Saharan
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Abstract

In this article, we study Riemannian planes (M,g) which satisfies a bounded geometry condition ( bounded sectional curvature and a non-vanishing injectivity radius of (M,g)). We show that such a Riemannian plane is always quasi-isometric to a radially symmetric plane which also satisfies the aforesaid bounded geometry condition. Using that, we study the quasi-redirecting boundary of Riemannian planes with bounded geometry and show that it is either a single point or π•Š1. We contrast the quasi-redirecting boundary with the Martin boundary and construct Riemannian planes with bounded geometry for which the We also characterise quasi-redirection in Riemannian planes with bounded geometry via their asymptotic cones and their cut points. Lastly, we characterise 𝔼2 and ℍ2 upto quasi-isometry by their asymptotic cones.

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Cite

BibTeXRIS
BibTeX
@misc{saharan2026geodesic,
  title = {{Geodesic Divergence on Riemannian Planes with Bounded Geometry}},
  author = {R Saharan},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2606.27393},
  eprint = {2606.27393},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.27393},
}
RIS
TY  - UNPB
TI  - Geodesic Divergence on Riemannian Planes with Bounded Geometry
AU  - R Saharan
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2606.27393
UR  - https://arxiv.org/abs/2606.27393
AB  - In this article, we study Riemannian planes (M,g) which satisfies a bounded geometry condition ( bounded sectional curvature and a non-vanishing injectivity radius of (M,g)). We show that such a Riemannian plane is always quasi-isometric to a radially symmetric plane which also satisfies the aforesaid bounded geometry condition. Using that, we study the quasi-redirecting boundary of Riemannian planes with bounded geometry and show that it is either a single point or π•Š^1. We contrast the quasi-redirecting boundary with the Martin boundary and construct Riemannian planes with bounded geometry for which the We also characterise quasi-redirection in Riemannian planes with bounded geometry via their asymptotic cones and their cut points. Lastly, we characterise 𝔼^2 and ℍ^2 upto quasi-isometry by their asymptotic cones.
ER  -