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

Mateo Anarella, Joeri Van der Veken

Abstract

In this paper, we study Lagrangian submanifolds of the pseudo-nearly Kähler $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$. 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.

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

Abstract

In this paper, we study Lagrangian submanifolds of the pseudo-nearly Kähler . 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.
Paper Structure (13 sections, 18 theorems, 107 equations, 4 tables)

This paper contains 13 sections, 18 theorems, 107 equations, 4 tables.

Table of Contents

  1. Introduction
  2. The pseudo-nearly Kähler SL(2,R)xSl(2,R)
  3. Nearly Kähler manifolds
  4. The manifold SL(2,R)
  5. The nearly Kähler structure on SL(2,R)xSL(2,R)
  6. The almost product structure P
  7. The isometries of SL(2,R)xSL(2,R)
  8. Comparison with the product metric
  9. Lagrangian submanifolds of SL(2,R)xSL(2,R)
  10. Totally geodesic Lagrangian submanifolds
  11. Lagrangian submanifolds of diagonalizable type
  12. Lagrangian submanifolds of the non-diagonalizable types
  13. Proof of the main result

Key Result

Theorem 1

Any totally geodesic Lagrangian submanifold of the pseudo-nearly Kähler $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ is congruent to the image of one of the following maps, possibly restricted to an open subset: where $\operatorname{Id}_2,\textit{i},\textbf{k}$ are the matrices Conversely, the maps (map1), (map2) and (map3) are totally geodesic Lagrangian immersions.

Theorems & Definitions (37)

  • Theorem 1
  • Proposition 2: Proposition 3.2 in Ghandour
  • Proposition 3
  • Lemma 4
  • proof
  • Remark
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 27 more