Umbilical routes along geodesics and hypercycles in the hyperbolic space

TL;DR

This paper establishes conditions for functions measuring mean curvature of umbilical leaves in hyperbolic space along geodesics and hypercycles, extending geometric understanding of foliations orthogonal to these curves.

Contribution

It provides a necessary and sufficient condition for such functions along geodesics and hypercycles in hyperbolic space, extending previous geometric results.

Findings

Derived conditions for mean curvature functions along geodesics

Extended results to hypercycles in hyperbolic space

Enhanced understanding of foliations orthogonal to geodesics and hypercycles

Abstract

Given a geodesic line $\gamma$ the hyperbolic space $\mathbb H^n$ we formulate a necessary and sufficient condition for a function along this geodesic which measure the mean curvature of totally umbilical leaves of a foliation orthogonal to $\gamma$. Then we extend the result to $\gamma$ being a hypercycle i.e. a geodesic on a hypersurface equidistant from the totally geodesic one.