Year: 2026
Type: preprint
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 $\mathbb{S}^1$. 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 $\mathbb{E}^2$ and $\mathbb{H}^2$ upto quasi-isometry by their asymptotic cones.
Source: arXiv (Cornell University)
Author: Saharan, R
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166571547