Geodesic Divergence on Riemannian Planes with Bounded Geometry

Abstract

In this article, we study Riemannian planes $(M,g)$ which satisfies a certain bounded geometry condition and geodesic divergence on these Riemannian planes. We recall quasi-redirection (introduced by Qing and Rafi) which generalises Gromov's bordification for $δ-$hyperbolic spaces and use it to quantify geodesic divergence in a manner which is invariant under quasi-isometries. We use it to compactify Riemannian planes into either $\mathbb{D}^2$ or $\mathbb{S}^2$ depending on how fast geodesics on $(M,g)$ spread apart. We study asymptotic cones of Riemannian planes and use them to come up with necessary and sufficient conditions for the quasi-redirecting compactification being $\mathbb{S}^2$ in terms of it admitting a proper asymptotic cone . Lastly, we study the Martin boundary of Riemannian planes ( with respect to the Laplace-Beltrami operator $Δ_g$) in relation to the quasi-redirecting boundary and show that if the quasi-redirecting boundary is homeomorphic to the Martin boundary, then the identity map on $(M,g)$ induces a homeomorphism from the quasi-redirecting compactification of $(M,g)$ to the Martin compactification of $(M,g)$.