Almost complex totally geodesic surfaces in the nearly Kähler $\frac{\text{SL}(3,\mathbb R)}{\mathbb R\times \text{SO}(2)}$

TL;DR

This work classifies totally geodesic almost complex surfaces in the pseudo-Roinn nearly Kähler space $\mathrm{SL}(3,\mathbb{R})/\big(\mathbb{R}\times\mathrm{SO}(2)\big)$. It constructs the homogeneous structure with a reductive decomposition, invariant metric, and two invariant almost complex structures $(J,J_1)$ (with $J_1$ integrable) plus an $F$-structure, and derives an explicit curvature formula. The authors identify five explicit examples, realized as orbits of subgroups $\mathrm{SL}(2,\mathbb{R})$, $\mathrm{SO}(3)$, $\mathrm{SO}^+(2,1)$, and $\mathbb{R}^2$ (including degenerate variants), which exhaust all totally geodesic almost complex surfaces up to local congruence. The classification hinges on analyzing possible tangent directions via the $\mathrm{Ad}(H)$-invariant decomposition and curvature preservation, establishing a complete local picture in this pseudo-/Riemannian setting. This extends known classifications from flag-manifold analogues to the six-dimensional, non-Riemannian context, contributing to the broader understanding of nearly Kähler geometries.

Abstract

We give a detailed description of the nearly Kähler $\frac{\mathrm{SL}(3,\mathbb R)}{\mathbb R\times \mathrm{SO}(2)}$, which is one of the pseudo-Riemannian counterparts of the flag manifold $F(\mathbb{C}^3)$. The main result is the classification of totally geodesic almost complex surfaces in this space.

Theorems & Definitions (12)

• Theorem A
• Definition 1.1
• Definition 1.2
• Lemma 1.1
• proof
• Lemma 3.1
• Lemma 3.2
• Example 3.1
• Example 3.2
• Example 3.3