Loxodromes on invariant surface in three-manifolds

TL;DR

This paper studies loxodromes on invariant surfaces within three-dimensional Riemannian manifolds, generalizing classical results and providing explicit parametrizations in hyperbolic spaces and minimal surfaces.

Contribution

It introduces new parametrizations of loxodromes on invariant surfaces in hyperbolic spaces and minimal surfaces, extending classical Euclidean results to more general manifolds.

Findings

Parametrization of loxodromes on invariant surfaces in    and    spaces.

Explicit descriptions of loxodromes on constant Gauss curvature surfaces.

Generalization of classical loxodrome results to non-Euclidean manifolds.

Abstract

In this paper, we prove important results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, some of which generalize classical results about loxodromes on rotational surfaces in $\mathbb{R}^3$. In particular, we show how to parametrize a loxodrome on an invariant surface of $\mathbb{H}^2\times \mathbb{R}$ and $\mathbb{H}_3$ and we exhibit the loxodromes of some remarkable minimal invariant surfaces of these spaces. In addition, we give an explicit description of the loxodromes on an invariant surface with constant Gauss curvature.