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On a class of hypersurfaces of a product of two space forms

Arnando Carvalho, Ruy Tojeiro

Abstract

We define hypersurfaces $f\colon M^n\to \mathbb{Q}_{c_1}^{k} \times \mathbb{Q}_{c_2}^{n-k+1}$ in class $\mathcal{A}$ of a product of two space forms as those that have flat normal bundle when regarded as submanifolds of the underlying flat ambient space. We provide an explicit construction of them in terms of parallel families of hypersurfaces of the factors. We show that hypersurfaces with constant mean curvature in class $\mathcal{A}$ are given in terms of parallel families of isoparametric hypersurfaces in each factor and a solution of a second order ODE. Finally, we classify hypersurfaces with constant mean curvature in class $\mathcal{A}$ that have constant product angle function.

On a class of hypersurfaces of a product of two space forms

Abstract

We define hypersurfaces in class of a product of two space forms as those that have flat normal bundle when regarded as submanifolds of the underlying flat ambient space. We provide an explicit construction of them in terms of parallel families of hypersurfaces of the factors. We show that hypersurfaces with constant mean curvature in class are given in terms of parallel families of isoparametric hypersurfaces in each factor and a solution of a second order ODE. Finally, we classify hypersurfaces with constant mean curvature in class that have constant product angle function.
Paper Structure (6 sections, 7 theorems, 65 equations)

This paper contains 6 sections, 7 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypersurfaces in class $\mathcal{A}$
  4. Constant mean curvature hypersurfaces of $\mathbb{Q}_{c_1}^{k} \times \mathbb{Q}_{c_2}^{n-k+1}$ in class $\mathcal{A}.$
  5. Hypersurfaces of $\mathbb{Q}_{c_1}^{k} \times \mathbb{Q}_{c_2}^{n-k+1}$ in class $\mathcal{A}$ with constant product angle function
  6. CMC hypersurfaces of $\mathbb{Q}_{c_1}^{k} \times \mathbb{Q}_{c_2}^{n-k+1}$ in class $\mathcal{A}$ with constant product angle function

Key Result

Proposition 1

A hypersurface $f\colon M^{n}\rightarrow \mathbb{Q}_{c_1}^{k}\times \mathbb{Q}_{c_2}^{n-k+1}$ splits locally if and only if $\xi_{f}$ vanishes and neither $R=0$ nor $R=I$ if $k=n-1$ or $k=1$, respectively.

Theorems & Definitions (13)

  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Example 7
  • Example 8
  • Example 9
  • Example 10
  • ...and 3 more