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Classification results for totally real surfaces of nearly Kähler $\mathbb{C}P^3$

Michaël Liefsoens, Hui Ma, Luc Vrancken

TL;DR

We classify totally real surfaces in the six-dimensional nearly Kähler manifold $(\mathbb{C}P^3, g, J)$ under several geometric constraints, constructing explicit models and proving rigidity in key cases. The approach combines a canonical adapted frame, the Hopf-twistor description, and Legendre curve data to generate and analyse examples, including a complete extrinsically homogeneous classification and a broad Codazzi-like family arising from products of Legendre curves in $\mathbb{S}^3$ and $\mathbb{S}^7$. We identify two principal construction paradigms: (i) flat-torus and sphere families from group orbits, and (ii) Codazzi-type surfaces given by $F(u,v)=f(v)\gamma(u)$ with $\gamma$ Legendre; these yield all parallel and many minimal or totally geodesic instances, with precise characterisations relative to the ambient Kähler and Lagrangian geometries. The results deepen the submanifold theory in nearly Kähler $\mathbb{C}P^3$, connect to Nagy’s structure theorem and homogeneous classifications, and provide explicit geometric models and a framework guiding future existence/uniqueness investigations.

Abstract

Totally real surfaces in the nearly Kähler $\mathbb{C}P^3$ are investigated and are completely classified under various additional assumptions, resulting in multiple new examples. Among others, the classification includes totally real surfaces that are extrinsically homogeneous; or minimal; or totally umbilical; or Codazzi-like (including parallel and non-parallel examples).

Classification results for totally real surfaces of nearly Kähler $\mathbb{C}P^3$

TL;DR

We classify totally real surfaces in the six-dimensional nearly Kähler manifold under several geometric constraints, constructing explicit models and proving rigidity in key cases. The approach combines a canonical adapted frame, the Hopf-twistor description, and Legendre curve data to generate and analyse examples, including a complete extrinsically homogeneous classification and a broad Codazzi-like family arising from products of Legendre curves in and . We identify two principal construction paradigms: (i) flat-torus and sphere families from group orbits, and (ii) Codazzi-type surfaces given by with Legendre; these yield all parallel and many minimal or totally geodesic instances, with precise characterisations relative to the ambient Kähler and Lagrangian geometries. The results deepen the submanifold theory in nearly Kähler , connect to Nagy’s structure theorem and homogeneous classifications, and provide explicit geometric models and a framework guiding future existence/uniqueness investigations.

Abstract

Totally real surfaces in the nearly Kähler are investigated and are completely classified under various additional assumptions, resulting in multiple new examples. Among others, the classification includes totally real surfaces that are extrinsically homogeneous; or minimal; or totally umbilical; or Codazzi-like (including parallel and non-parallel examples).

Paper Structure

This paper contains 22 sections, 13 theorems, 53 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Description of the nearly Kähler ℂP³
  3. Nearly Kähler ℂP³ via the Hopf fibration
  4. Nearly Kähler ℂP³ as a homogeneous space
  5. A frame for totally real surfaces
  6. Intermezzo: Legendre curves in spheres
  7. Legendre curves in S³ ⊂ ℂ²
  8. Legendre curves in S⁷ ⊂ ℍ²
  9. Examples of totally real surfaces
  10. Extrinsically homogeneous examples
  11. Family 1
  12. Family 2
  13. Family 3
  14. Family 4
  15. Sphere
  16. ...and 7 more sections

Key Result

Lemma 3.1

Let $h$ denote the second fundamental form of a totally real surface in a nearly Kähler manifold $(M, g, J)$. Then, $(X,Y,Z) \mapsto g(h(X,Z), JY )$ is totally symmetric for all $X,Y,Z$ tangent to $M$.

Figures (1)

  • Figure 1: Up to $\mathop{\mathrm{Ad}}\nolimits_g: \mathfrak{sp}(2) \to \mathfrak{sp}(2): X \mapsto g X g^{-1}$ for some $g \in \mathop{\mathrm{Sp}}\nolimits(2)$, this diagram gives all Lie sub algebras of $\mathfrak{sp}(2)$. Modified from kerr2013, where we used $\mathfrak{so}(5) = \mathfrak{sp}(2)$. Note that the chain $\mathop{\mathrm{SU}}\nolimits(2) \mathop{\mathrm{U}}\nolimits(1) \to \mathop{\mathrm{SU}}\nolimits(2)\times\mathop{\mathrm{SU}}\nolimits(2) \to \mathop{\mathrm{Sp}}\nolimits(2)$ (whose corresponding Lie algebra sequence is indicated with orange dotted arrows) gives rise to the twistor fibration $\uptau: {\mathbb{C} P}^3 \to \mathbb{S}^{4}$. Moreover, the chain $\mathop{\mathrm{SU}}\nolimits(2) \to \mathop{\mathrm{SU}}\nolimits(2)\times\mathop{\mathrm{U}}\nolimits(1) \to \mathop{\mathrm{Sp}}\nolimits(2)$ (corresponding sequence is indicated with blue dashed arrows) gives rise to the Hopf fibration $\uppi: \mathbb{S}^{7} \to {\mathbb{C} P}^3$.

Theorems & Definitions (27)

  • Conjecture 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 1
  • Proposition 3.3
  • proof
  • Theorem 6.1
  • proof
  • ...and 17 more