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Isoparametric Hypersurfaces in Products of Simply Connected Space Forms

Ronaldo F. de Lima, Giuseppe Pipoli

TL;DR

This work establishes that every connected isoparametric hypersurface in a product of simply connected space forms $\mathbb{Q}_{\varepsilon_1}^{n_1}\times\mathbb{Q}_{\varepsilon_2}^{n_2}$ with $\varepsilon_1^2+\varepsilon_2^2\neq 0$ has a constant angle function $\Theta$ between the ambient product structure and the normal. Using Jacobi-field methods, the authors reduce the isoparametric condition to an algebraic system whose solutions force $\Theta$ to be constant, enabling a classification of isoparametric and homogeneous hypersurfaces under a one-point condition. The classification yields that such hypersurfaces are open sets of either products of isoparametric hypersurfaces in each factor, flat-horospherical, or bi-horospherical types, with homogeneity tied to the signs of the curvatures when present. The paper also develops a detailed algebraic apparatus based on modified Kac matrices to handle the Jacobi-field equations, including case distinctions when both factors have nonzero curvature or one factor is flat. The results extend known classifications in particular cases (e.g., $\mathbb{H}^n$, $\mathbb{S}^n$, cylinders) and provide a framework for further study of constant-angle hypersurfaces in product space forms. The approach highlights the deep interplay between geometric structure and linear-algebraic constraints in higher-rank product spaces.

Abstract

Let $\mathbb Q_{ε_i}^{n_i}$ denote the simply connected space form of dimension $n_i\ge 2$ and constant sectional curvature $ε_i$. We prove that any connected isoparametric hypersurface of $\mathbb Q_{ε_1}^{n_1}\times\mathbb Q_{ε_2}^{n_2}$ has constant angle function. We then use this property to classify the isoparametric and homogeneous hypersurfaces of $\mathbb Q_{ε_1}^{n_1}\times\mathbb Q_{ε_2}^{n_2}$, $|ε_1|+|ε_2|\ne 0$, that satisfy a one-point condition.

Isoparametric Hypersurfaces in Products of Simply Connected Space Forms

TL;DR

This work establishes that every connected isoparametric hypersurface in a product of simply connected space forms with has a constant angle function between the ambient product structure and the normal. Using Jacobi-field methods, the authors reduce the isoparametric condition to an algebraic system whose solutions force to be constant, enabling a classification of isoparametric and homogeneous hypersurfaces under a one-point condition. The classification yields that such hypersurfaces are open sets of either products of isoparametric hypersurfaces in each factor, flat-horospherical, or bi-horospherical types, with homogeneity tied to the signs of the curvatures when present. The paper also develops a detailed algebraic apparatus based on modified Kac matrices to handle the Jacobi-field equations, including case distinctions when both factors have nonzero curvature or one factor is flat. The results extend known classifications in particular cases (e.g., , , cylinders) and provide a framework for further study of constant-angle hypersurfaces in product space forms. The approach highlights the deep interplay between geometric structure and linear-algebraic constraints in higher-rank product spaces.

Abstract

Let denote the simply connected space form of dimension and constant sectional curvature . We prove that any connected isoparametric hypersurface of has constant angle function. We then use this property to classify the isoparametric and homogeneous hypersurfaces of , , that satisfy a one-point condition.

Paper Structure

This paper contains 11 sections, 22 theorems, 201 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some geometric aspects of $\mathbb{Q} _{\epsilon_1}^{n_1}\times\mathbb{Q} _{\epsilon_2}^{n_2}$
  4. Hypersurfaces of $\mathbb{Q} _{\epsilon_1}^{n_1}\times\mathbb{Q} _{\epsilon_2}^{n_2}$
  5. Hypersurfaces of $\mathbb{Q} _{\epsilon_1}^{n_1}\times\mathbb{Q} _{\epsilon_2}^{n_2}$ with constant angle
  6. Proofs of Theorems \ref{['thm-isoparametric-->thetaconstant']}--\ref{['thm-classification01']}
  7. Appendix
  8. Modified Kac matrices
  9. The case $\epsilon_1\epsilon_2\ne 0$
  10. The case $\epsilon_1=0\ne\epsilon_2$
  11. Final lemma

Key Result

Theorem 1

Every connected isoparametric hypersurface of $\mathbb{Q} _{\epsilon_1}^{n_1}\times\mathbb{Q} _{\epsilon_2}^{n_2}$, $\epsilon_1^2+\epsilon_2^2\ne0$, has constant angle function.

Figures (1)

  • Figure 1: Coordinates on a flat-horospherical hypersurface $\varSigma=\Phi(M_1\times M_2\times\mathbb{R} )\subset\mathbb{R} ^{n_1}\times\mathbb{H}^{n_2}_{\epsilon_2}$, $\Phi(p_1,p_2, s):=(f_s^1(p_1), f_{as}^2(p_2))$, where $a>0$ and $\{f_s^1\}$ (resp. $\{f_{as}^2\}$) is a parallel family of hyperplanes of $\mathbb{R} ^{n_1}$ (resp. horospheres of $\mathbb{H}^{n_2}_{\epsilon_2}$); see Ex. \ref{['examp-horospherical']}, Sect. \ref{['sec-constantangle']}.

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Lemma 4
  • Definition 5
  • Remark 6
  • Lemma 7
  • proof
  • Definition 8
  • Definition 9
  • ...and 42 more