Totally geodesic Lagrangian submanifolds of the pseudo-nearly K\"ahler $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$

TL;DR

This paper classifies totally geodesic Lagrangian submanifolds within the pseudo-nearly Kähler space formed by the product of two SL(2,R) groups, revealing four distinct classes based on an almost product structure.

Contribution

It provides a complete classification of totally geodesic Lagrangian submanifolds in the specific pseudo-nearly Kähler manifold, highlighting four classes based on their behavior.

Findings

Lagrangian submanifolds split into four classes

Complete classification of totally geodesic Lagrangian submanifolds

Identification of behavior with respect to an almost product structure

Abstract

In this paper, we study Lagrangian submanifolds of the pseudo-nearly K\"ahler $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$. First, we show that they split into four different classes depending on their behaviour with respect to a certain almost product structure on the ambient space. Then, we give a complete classification of totally geodesic Lagrangian submanifolds of this space.