Almost complex surfaces in the nearly Kahler SL(2,R)xSL(2,R)

TL;DR

This paper classifies almost complex surfaces in the nearly Kaehler space SL(2,R)×SL(2,R), focusing on totally geodesic and parallel second fundamental form cases, revealing their geometric properties.

Contribution

It provides a complete classification of specific almost complex surfaces in a homogeneous nearly Kaehler manifold, a novel result in differential geometry.

Findings

Classification of totally geodesic almost complex surfaces

Classification of almost complex surfaces with parallel second fundamental form

Identification of geometric properties of these surfaces

Abstract

The space $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ admits a natural homogeneous pseudo-Riemannian nearly Kaehler structure. We investigate almost complex surfaces in this space. In particular we obtain a complete classification of the totally geodesic almost complex surfaces and of the almost complex surfaces with parallel second fundamental form.