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On Hopf hypersurfaces of the complex hyperbolic quadric with constant principal curvatures

Haizhong Li, Hiroshi Tamaru, Zeke Yao

TL;DR

This work classifies Hopf real hypersurfaces in the complex hyperbolic quadric $Q^{m\*}$ ($m\ge3$) with constant principal curvatures. It exploits the ambient two-structure geometry $(A,J)$ with $AJ=-JA$, separating cases by the normal type: $\mathfrak{A}$-principal and $\mathfrak{A}$-isotropic, and proves that such hypersurfaces are isoparametric with parallel families of constant principal curvatures. For at most two distinct curvatures, the hypersurface must be an open part of Example 3.1; for three or four distinct curvatures, the paper determines exact curvature values and multiplicities, aligning all canonical examples (Exs. 3.1–3.10) as homogeneous models. The analysis uses Jacobi-field methods for parallel hypersurfaces, Cartan-type formulas adapted to the almost product structure, and focal submanifold Austereness results, yielding a comprehensive picture of the Hopf hypersurface landscape in $Q^{m\*}$ and extending classifications beyond the complex hyperbolic space to this noncompact symmetric space.

Abstract

In this paper, we study the Hopf hypersurfaces of the complex hyperbolic quadric $Q^{m*}=SO^o_{2,m}/(SO_2\times SO_m)$ ($m\geq3$) with constant principal curvatures. We classify the Hopf hypersurfaces of $Q^{m*}$ ($m\geq3$) with at most two distinct constant principal curvatures. For Hopf hypersurfaces with three or four distinct constant principal curvatures, we determine the values of the principal curvatures as well as their multiplicities.

On Hopf hypersurfaces of the complex hyperbolic quadric with constant principal curvatures

TL;DR

This work classifies Hopf real hypersurfaces in the complex hyperbolic quadric () with constant principal curvatures. It exploits the ambient two-structure geometry with , separating cases by the normal type: -principal and -isotropic, and proves that such hypersurfaces are isoparametric with parallel families of constant principal curvatures. For at most two distinct curvatures, the hypersurface must be an open part of Example 3.1; for three or four distinct curvatures, the paper determines exact curvature values and multiplicities, aligning all canonical examples (Exs. 3.1–3.10) as homogeneous models. The analysis uses Jacobi-field methods for parallel hypersurfaces, Cartan-type formulas adapted to the almost product structure, and focal submanifold Austereness results, yielding a comprehensive picture of the Hopf hypersurface landscape in and extending classifications beyond the complex hyperbolic space to this noncompact symmetric space.

Abstract

In this paper, we study the Hopf hypersurfaces of the complex hyperbolic quadric () with constant principal curvatures. We classify the Hopf hypersurfaces of () with at most two distinct constant principal curvatures. For Hopf hypersurfaces with three or four distinct constant principal curvatures, we determine the values of the principal curvatures as well as their multiplicities.
Paper Structure (12 sections, 19 theorems, 111 equations)

This paper contains 12 sections, 19 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The complex hyperbolic quadric
  4. Real hypersurfaces of the complex hyperbolic quadric
  5. Canonical examples of Hopf hypersurfaces with constant principal curvatures
  6. Parallel hypersurfaces, focal submanifolds and Cartan's formulas
  7. Parallel hypersurfaces and focal submanifold of the Hopf hypersurface with constant principal curvatures and $\mathfrak{A}$-isotropic unit normal vector field
  8. Cartan's formulas for Hopf hypersurfaces with constant principal curvatures and $\mathfrak{A}$-isotropic unit normal vector field
  9. Proofs of Theorems \ref{['thm:1.1a']}, \ref{['thm:1.1b']} and \ref{['thm:1.1c']}
  10. The proof of Theorem \ref{['thm:1.1a']}
  11. The proof of Theorem \ref{['thm:1.1b']}
  12. The proof of Theorem \ref{['thm:1.1c']}

Key Result

Theorem 1.1

Let $M$ be a Hopf hypersurface of $Q^{m*}$ ($m\geq3$) with at most two distinct constant principal curvatures. Then $M$ is an open part of Example ex:3.1.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1: cf. BS-2022K-SM-RV-W
  • Definition 2.2: BS-2022Suh4
  • Remark 2.1
  • Lemma 2.3: BS-2022
  • ...and 38 more