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

Mateo Anarella

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.

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 and we study its Lagrangian submanifolds. We provide examples of Lagrangian submanifolds which do not have an analogue in . We also provide an expression for the isometry group of with the pseudo-Riemannian nearly Kähler metric. The main result is a complete classification of extrinsically homogeneous Lagrangian submanifolds in this space.
Paper Structure (22 sections, 25 theorems, 180 equations, 1 table)

This paper contains 22 sections, 25 theorems, 180 equations, 1 table.

Table of Contents

  1. Introduction
  2. The pseudo-nearly Kähler SL(2,R)xSl(2,R)
  3. The manifold SL(2,R)
  4. The homogeneous nearly Kähler structure on SL(2,R)xSL(2,R)
  5. The manifold SL(2,R)xSL(2,R) as a pseudo-Riemannian product
  6. The isometry group
  7. Lagrangian submanifolds of SL(2,R)xSL(2,R)
  8. Lagrangian submanifolds of type I
  9. Lagrangian submanifolds of type II
  10. Lagrangian submanifolds of type III
  11. Lagrangian submanifolds of type IV
  12. Extrinsically homogeneous Lagrangian submanifolds
  13. The uniqueness of the frames
  14. Lagrangian submanifolds of type I
  15. Lagrangian submanifolds of type II
  16. ...and 7 more sections

Key Result

Theorem 1

The isometry group of the pseudo-nearly Kähler $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ is $(\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R}))\rtimes(\mathbb{Z}_2\times S_3)$, where $S_3$ is the symmetric group of order 6.

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem \ref{groupofisometries1}
  • Remark 5
  • proof : Proof of Theorem \ref{['groupofisometries1']}
  • Proposition 6
  • Lemma 7: anarella
  • ...and 39 more