Canonical Coordinates and Principal Directions for Surfaces in $\H^2\times\R$

TL;DR

This paper classifies and characterizes surfaces in hyperbolic plane times real line with a canonical principal direction, exploring their geometric properties like minimality and flatness, and providing illustrative examples.

Contribution

It introduces a classification of surfaces with a canonical principal direction in ${ m H}^2 imes m R$, including geometric properties and explicit examples.

Findings

Identification of conditions for minimal surfaces

Characterization of flat surfaces in ${ m H}^2 imes m R$

Construction of explicit examples of such surfaces

Abstract

In this paper we characterize and classify surfaces in ${\mathbb{H}}^2\times{\mathbb{R}}$ which have a canonical principal direction. Here ${\mathbb{H}}^2$ denotes the hyperbolic plane. We study some geometric properties such as minimality and flatness. Some examples are given to complete the study.