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Isoparametric Hypersurfaces of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R$

Ronaldo F. de Lima, Giuseppe Pipoli

TL;DR

The paper classifies isoparametric and homogeneous hypersurfaces in the product spaces $\mathbb Q_\epsilon^n\times\mathbb R$ ($n\ge 2$) by proving that such hypersurfaces must have constant angle function $\Theta$ and constant principal curvatures. The authors employ Jacobi-field theory to reduce the constant-angle condition to algebraic constraints on a parameter $\tau$, leading to a complete classification: horizontal slices, vertical cylinders over complete isoparametric hypersurfaces, and a parabolic bowl in the hyperbolic case, with homogeneous characterizations in $\mathbb H^n\times\mathbb R$. They further connect isoparametric hypersurfaces to constant scalar curvature and relate the results to geometric flows and existing classifications in space forms. The work leverages a detailed linear-algebra framework (in the appendix) to obtain the requisite algebraic conditions, establishing a rigorous foundation for the constant-angle phenomenon in these product manifolds.

Abstract

We classify the isoparametric hypersurfaces and the homogeneous hypersurfaces of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R$, $n\ge 2$, by establishing that any such hypersurface has constant angle function and constant principal curvatures.

Isoparametric Hypersurfaces of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R$

TL;DR

The paper classifies isoparametric and homogeneous hypersurfaces in the product spaces () by proving that such hypersurfaces must have constant angle function and constant principal curvatures. The authors employ Jacobi-field theory to reduce the constant-angle condition to algebraic constraints on a parameter , leading to a complete classification: horizontal slices, vertical cylinders over complete isoparametric hypersurfaces, and a parabolic bowl in the hyperbolic case, with homogeneous characterizations in . They further connect isoparametric hypersurfaces to constant scalar curvature and relate the results to geometric flows and existing classifications in space forms. The work leverages a detailed linear-algebra framework (in the appendix) to obtain the requisite algebraic conditions, establishing a rigorous foundation for the constant-angle phenomenon in these product manifolds.

Abstract

We classify the isoparametric hypersurfaces and the homogeneous hypersurfaces of and , , by establishing that any such hypersurface has constant angle function and constant principal curvatures.

Paper Structure

This paper contains 9 sections, 16 theorems, 113 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor Curvature of $\mathbb{Q} _\epsilon^n\times\mathbb{R}$
  4. Hypersurfaces of $\mathbb{Q} _\epsilon^n\times\mathbb{R}$
  5. Graphs on parallel hypersurfaces
  6. Isoparametric hypersurfaces of constant angle
  7. The constancy of the angle function
  8. Proof of Theorem \ref{['thm-main']}
  9. Appendix

Key Result

Theorem 1

Let $\varSigma$ be a connected hypersurface of $\mathbb Q_\epsilon^n\times\mathbb{R}$. Then the following are equivalent: Moreover, in the hyperbolic case $\epsilon=-1$, the condition is also equivalent to (i)--(iii). In the spherical case $\epsilon=1$, (iv) is equivalent to

Figures (1)

  • Figure 1: The depicted hypersurface, called a parabolic bowl, is a homogeneous entire vertical graph in $\mathbb{H}^n\times\mathbb{R}$ whose level hypersurfaces are parallel horospheres, and whose vertical translations define a nonsingular isoparametric foliation of $\mathbb{H}^n\times\mathbb{R}$ (we are grateful to João P. dos Santos for this plot).

Theorems & Definitions (33)

  • Theorem 1
  • Definition 1
  • Lemma 2: Theorem 7 of delima-roitman
  • Lemma 3: Corollary 4 of delima-roitman
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Remark 6
  • Proposition 7
  • ...and 23 more