Degenerate almost complex surfaces in the nearly K\"ahler $\mathrm{SL}_2\mathbb{R}\times \mathrm{SL}_2\mathbb{R}$

TL;DR

This paper classifies degenerate almost complex surfaces in the semi-Riemannian nearly Kähler space $ ext{SL}_2 ext{R} imes ext{SL}_2 ext{R}$, revealing how their geometry depends on the ambient space's almost product structure.

Contribution

We provide a complete and explicit classification of degenerate almost complex surfaces in the nearly Kähler space, considering cases based on the invariance of the tangent bundle under the almost product structure.

Findings

Classification of surfaces when tangent bundle is preserved

Classification of surfaces when tangent bundle is not preserved

Explicit geometric descriptions in both cases

Abstract

In this paper, we study degenerate almost complex surfaces in the semi-Riemannian nearly K\"ahler $\mathrm{SL}_2\mathbb{R}\times \mathrm{SL}_2\mathbb{R}$. The geometry of these surfaces depends on the almost product structure of the ambient space and one can distinguish two distinct cases. The geometry of these surfaces is influenced by the almost product structure of the ambient space, leading to two distinct cases. The first case arises when the tangent bundle of the surface is preserved under the almost product structure, while the second case occurs when the tangent bundle of the surface is not invariant under this structure. In both cases, we obtain a complete and explicit classification.