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Curvature and Lagrangian submanifolds of the homogeneous nearly Kähler $\mathbb{C}P^3$

Michaël Liefsoens, Joeri Van der Veken

TL;DR

This work provides a tractable Hopf-fibration model of the homogeneous nearly Kähler structure on ${\mathbb{C}P}^3$, yielding explicit curvature expressions for the entire family of homogeneous metrics ${g}_a$ and a complete description of their isometry groups. It then executes a detailed study of Lagrangian submanifolds in this setting, introducing an intrinsic angle function and giving geometrically transparent constructions for the main extrinsically homogeneous examples, along with a rigidity result: these examples are precisely the extrinsically homogeneous Lagrangians, up to isometries. The paper also proves a nonexistence result for Lagrangians with constant sectional curvature, highlighting the restrictive nature of the NK geometry in ${\mathbb{C}P^3}$. Overall, it advances both the global metric picture and the submanifold theory in homogeneous nearly Kähler ${\mathbb{C}P^3}$, with implications for rigidity and symmetry in related geometric structures.

Abstract

A tractable definition of the homogeneous nearly Kähler structure on $\mathbb{C}P^3$ is given via the Hopf fibration, facilitating explicit computations and analysis. The description extends to all homogeneous metrics on $\mathbb{C}P^3$, providing expressions for their Riemann curvature tensors and full isometry groups. Rigid immersions are presented for all extrinsically homogeneous Lagrangian submanifolds in the nearly Kähler $\mathbb{C}P^3$, and the nonexistence of Lagrangians with constant sectional curvature is established.

Curvature and Lagrangian submanifolds of the homogeneous nearly Kähler $\mathbb{C}P^3$

TL;DR

This work provides a tractable Hopf-fibration model of the homogeneous nearly Kähler structure on , yielding explicit curvature expressions for the entire family of homogeneous metrics and a complete description of their isometry groups. It then executes a detailed study of Lagrangian submanifolds in this setting, introducing an intrinsic angle function and giving geometrically transparent constructions for the main extrinsically homogeneous examples, along with a rigidity result: these examples are precisely the extrinsically homogeneous Lagrangians, up to isometries. The paper also proves a nonexistence result for Lagrangians with constant sectional curvature, highlighting the restrictive nature of the NK geometry in . Overall, it advances both the global metric picture and the submanifold theory in homogeneous nearly Kähler , with implications for rigidity and symmetry in related geometric structures.

Abstract

A tractable definition of the homogeneous nearly Kähler structure on is given via the Hopf fibration, facilitating explicit computations and analysis. The description extends to all homogeneous metrics on , providing expressions for their Riemann curvature tensors and full isometry groups. Rigid immersions are presented for all extrinsically homogeneous Lagrangian submanifolds in the nearly Kähler , and the nonexistence of Lagrangians with constant sectional curvature is established.
Paper Structure (17 sections, 30 theorems, 66 equations)

This paper contains 17 sections, 30 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kähler ℂP³
  4. Nearly Kähler manifolds of constant type
  5. A congruence result
  6. Nearly Kähler ℂP³ via the Hopf fibration
  7. The almost product structure on ℂP³
  8. A contact frame on ℂP³
  9. The homogeneous nearly Kähler metric on ℂP³
  10. Curvature of (ℂP³,ga)
  11. Isometries of (ℂP³,ga)
  12. Lagrangian submanifolds of the nearly Kähler ℂP³
  13. General properties of Lagrangian submanifolds of nearly Kähler six-manifolds
  14. Definition of the angle function
  15. Examples of Lagrangian submanifolds of the nearly Kähler ℂP³
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $(M, g)$ and $(\tilde{M}, \tilde{g})$ be $n$-dimensional Riemannian manifolds with Levi-Civita connections $\nabla$ and $\tilde{\nabla}$. Suppose that there exist constants $c_{ij}^k$$(i,j,k\in\left\{ 1,\hdots,n \right\})$ such that for all $p\in M$ and $\tilde{p}\in \tilde{M}$ there exist local

Theorems & Definitions (66)

  • Proposition 2.1: dioos2018
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • Remark 1
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 56 more