Surfaces in a pseudo-sphere with harmonic or 1-type pseudo-spherical Gauss map

TL;DR

This paper classifies Riemannian and Lorentzian surfaces in pseudo-spheres based on their pseudo-spherical Gauss maps being of 1-type or harmonic, providing explicit solutions and global classifications.

Contribution

It offers a complete classification of such surfaces, including explicit solutions and global descriptions, for the first time in this context.

Findings

Classification of surfaces with harmonic or 1-type pseudo-spherical Gauss maps

Explicit PDE systems describing solutions in certain cases

Global classification results for specific surface types

Abstract

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification is obtained, while in other cases the solutions are described by an explicit system of partial differential equations.