Non-degenerate anisocurved surfaces in homogeneous 3-manifolds

TL;DR

This paper studies non-degenerate surfaces in homogeneous 3-manifolds with two different metrics, comparing their Gaussian curvatures, and characterizes anisocurved surfaces where these curvatures are opposite.

Contribution

It introduces the concept of anisocurved surfaces in homogeneous 3-manifolds and analyzes their properties under dual metric structures.

Findings

Characterization of anisocurved surfaces where Gaussian curvatures are opposite.

Comparison of surface geometry under Riemannian and Lorentzian metrics.

Conditions involving extrinsic curvatures for such surfaces.

Abstract

In this manuscript we consider non-degenerate surfaces $\Sigma^2$ immersed in a 3-dimensional homogeneous space $\mathbb{L}^3(\kappa,\tau)$ endowed with two different metrics, the one induced by the Riemannian metric of $\mathbb{E}^3(\kappa,\tau)$ and the non-degenerate metric inherited by the Lorentzian one of $\mathbb{L}^3(\kappa,\tau)$. Therefore, we have two different geometries on $\Sigma^2$ and we can compare them. In particular, we can consider the Gaussian curvature functions which respect to both metrics and study the geometry of the surfaces satisfying that both Gaussian curvature functions are opposite. We will call these surfaces anisocurved surfaces. In order to obtain our main results we also need to impose some extra assumptions regarding the extrinsic curvatures with respect to both metrics.