Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hypersurfaces of any homogeneous $\mathbb{C}P^3$

Michaël Liefsoens

TL;DR

This work provides a comprehensive geometry of hypersurfaces in homogeneous nearly Kähler ${\mathbb{C}P}^3$ across the entire spectrum of homogeneous metrics ${g_a}$. It unifies extrinsic-homogeneity, Hopf structure, and twistor-fibration geometry, proving that all extrinsically homogeneous hypersurfaces are congruent to a canonical Hopf family, and that all Hopf hypersurfaces in the nearly Kähler setting are horizontal and reside in explicit tube-type examples. It also establishes strong non-existence results: Codazzi-like, parallel, totally geodesic, totally umbilical, and constant sectional curvature hypersurfaces do not occur in any ${\mathbb{C}P^3},{g_a}$. The paper leverages the angle function, complex almost contact structure, and twistor-Hopf fibration to link isotropic and constant-angle hypersurfaces, yielding explicit classifications and a clear intrinsic-extrinsic dichotomy with symmetry about a central parameter. Together, these results give a complete picture of hypersurface geometry in homogeneous ${\mathbb{C}P^3}$ and its nearly Kähler limit, with concrete families and rigorous non-existence proofs that sharpen the landscape for related geometric problems.

Abstract

Hypersurfaces are studied and classified under multiple additional assumptions in any Riemannian homogeneous space $(\mathbb{C}P^3, g_a)$, including nearly Kähler $\mathbb{C}P^3$. Notably, all extrinsically homogeneous hypersurfaces are classified in all these spaces, with an explicit family of examples. Moreover, for nearly Kähler $\mathbb{C}P^3$, all Hopf hypersurfaces are classified. Finally, Codazzi-like hypersurfaces (and in particular parallel and totally geodesic hypersurfaces), totally umbilical hypersurfaces and constant sectional curvature hypersurfaces are proven to not exist in any homogeneous $\mathbb{C}P^3$.

Hypersurfaces of any homogeneous $\mathbb{C}P^3$

TL;DR

This work provides a comprehensive geometry of hypersurfaces in homogeneous nearly Kähler across the entire spectrum of homogeneous metrics . It unifies extrinsic-homogeneity, Hopf structure, and twistor-fibration geometry, proving that all extrinsically homogeneous hypersurfaces are congruent to a canonical Hopf family, and that all Hopf hypersurfaces in the nearly Kähler setting are horizontal and reside in explicit tube-type examples. It also establishes strong non-existence results: Codazzi-like, parallel, totally geodesic, totally umbilical, and constant sectional curvature hypersurfaces do not occur in any . The paper leverages the angle function, complex almost contact structure, and twistor-Hopf fibration to link isotropic and constant-angle hypersurfaces, yielding explicit classifications and a clear intrinsic-extrinsic dichotomy with symmetry about a central parameter. Together, these results give a complete picture of hypersurface geometry in homogeneous and its nearly Kähler limit, with concrete families and rigorous non-existence proofs that sharpen the landscape for related geometric problems.

Abstract

Hypersurfaces are studied and classified under multiple additional assumptions in any Riemannian homogeneous space , including nearly Kähler . Notably, all extrinsically homogeneous hypersurfaces are classified in all these spaces, with an explicit family of examples. Moreover, for nearly Kähler , all Hopf hypersurfaces are classified. Finally, Codazzi-like hypersurfaces (and in particular parallel and totally geodesic hypersurfaces), totally umbilical hypersurfaces and constant sectional curvature hypersurfaces are proven to not exist in any homogeneous .

Paper Structure

This paper contains 18 sections, 19 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamental equations of hypersurfaces
  4. Types of hypersurfaces
  5. (Nearly Kähler) ℂP³ via the Hopf fibration
  6. Complex almost contact structure on ℂP³
  7. Angle function for hypersurfaces
  8. Linking isotropic and constant angle hypersurfaces
  9. Examples of Hypersurfaces
  10. ℋₜ is extrinsically homogeneous
  11. ℋₜ is Hopf
  12. ℋₜ as a tube
  13. ℋₜ projects to geodesic 𝕊³ ⊂ 𝕊⁴
  14. ℋₜ: extrinsic and intrinsic differences
  15. Hopf hypersurfaces in the nearly Kähler ℂP³
  16. ...and 3 more sections

Key Result

Theorem A

Let $\mathcal{H}$ be a hypersurface in the nearly Kähler ${\mathbb{C} P}^3$ with unit normal $N$, and let $\pi:\mathbb{S}^{7}\to {\mathbb{C} P}^3$ be the Hopf fibration. The following are equivalent Let $\varphi$ be the contact structure induced on $\mathcal{H}$, cfr. eq:almost_contact_structure. Then, moreover, the following are equivalent

Theorems & Definitions (37)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 3.1
  • Remark 1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • ...and 27 more