Isometric Immersions of the Hyperbolic Plane into the Hyperbolic Space

TL;DR

This paper characterizes isometric immersions of the hyperbolic plane into hyperbolic 3-space using null-causal curves and describes ideal cones through their mean curvature behavior.

Contribution

It introduces a parametrization of hyperbolic plane immersions via null-causal curves and characterizes ideal cones by mean curvature, advancing geometric understanding.

Findings

Parametrization of immersions via null-causal curves

Characterization of ideal cones by mean curvature behavior

New insights into hyperbolic geometry immersions

Abstract

In this paper, we parametrize the space of isometric immersions of the hyperbolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of oriented geodesics. Moreover, we characterize "ideal cones" (i.e., cones whose vertices are on the ideal boundary) by behavior of their mean curvature.