Extrinsically homogeneous Lagrangian submanifolds of the pseudo-nearly Kähler $\mathrm{SL}(2,\mathbb{R})\times \mathrm{SL}(2,\mathbb{R})$

Abstract

We consider the pseudo-nearly Kähler $\mathrm{SL}(2,\mathbb{R})\times \mathrm{SL}(2,\mathbb{R})$ and we study its Lagrangian submanifolds. We provide examples of Lagrangian submanifolds which do not have an analogue in ${\mathbb{S}}^3\times {\mathbb{S}}^3$. We also provide an expression for the isometry group of $\mathrm{SL}(2,\mathbb{R})\times \mathrm{SL}(2,\mathbb{R})$ with the pseudo-Riemannian nearly Kähler metric. The main result is a complete classification of extrinsically homogeneous Lagrangian submanifolds in this space.

33 pages, comments are welcome!