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Sphere-like isoparametric hypersurfaces in Damek-Ricci spaces

Balázs Csikós, Márton Horváth

Abstract

Locally harmonic manifolds are Riemannian manifolds in which small geodesic spheres are isoparametric hypersurfaces, i.e., hypersurfaces whose nearby parallel hypersurfaces are of constant mean curvature. Flat and rank one symmetric spaces are examples of harmonic manifolds. Damek-Ricci spaces are non-compact harmonic manifolds, most of which are non-symmetric. Taking the limit of an "inflating" sphere through a point $p$ in a Damek-Ricci space as the center of the sphere runs out to infinity along a geodesic half-line $γ$ starting from $p$, we get a horosphere. Similarly to spheres, horospheres are also isoparametric hypersurfaces. In this paper, we define the sphere-like hypersurfaces obtained by "overinflating the horospheres" by pushing the center of the sphere beyond the point at infinity of $γ$ along a virtual prolongation of $γ$. They give a new family of isoparametric hypersurfaces in Damek-Ricci spaces connecting geodesic spheres to some of the isoparametric hypersurfaces constructed by J. C. Díaz-Ramos and M. Domínguez-Vázquez [arXiv:1111.0264] in Damek-Ricci spaces. We study the geometric properties of these isoparametric hypersurfaces, in particular their homogeneity and the totally geodesic condition for their focal varieties.

Sphere-like isoparametric hypersurfaces in Damek-Ricci spaces

Abstract

Locally harmonic manifolds are Riemannian manifolds in which small geodesic spheres are isoparametric hypersurfaces, i.e., hypersurfaces whose nearby parallel hypersurfaces are of constant mean curvature. Flat and rank one symmetric spaces are examples of harmonic manifolds. Damek-Ricci spaces are non-compact harmonic manifolds, most of which are non-symmetric. Taking the limit of an "inflating" sphere through a point in a Damek-Ricci space as the center of the sphere runs out to infinity along a geodesic half-line starting from , we get a horosphere. Similarly to spheres, horospheres are also isoparametric hypersurfaces. In this paper, we define the sphere-like hypersurfaces obtained by "overinflating the horospheres" by pushing the center of the sphere beyond the point at infinity of along a virtual prolongation of . They give a new family of isoparametric hypersurfaces in Damek-Ricci spaces connecting geodesic spheres to some of the isoparametric hypersurfaces constructed by J. C. Díaz-Ramos and M. Domínguez-Vázquez [arXiv:1111.0264] in Damek-Ricci spaces. We study the geometric properties of these isoparametric hypersurfaces, in particular their homogeneity and the totally geodesic condition for their focal varieties.
Paper Structure (12 sections, 33 theorems, 92 equations)

This paper contains 12 sections, 33 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Isoparametric functions
  3. Damek--Ricci spaces
  4. The J2-condition
  5. The half-space model of Damek--Ricci spaces
  6. Distance-like isoparametric functions in Damek--Ricci spaces
  7. The focal varieties of the functions D(x0)
  8. Prolongation of the family of spheres tangent to one another at one point
  9. Geodesic curves orthogonal to a focal variety
  10. Totally geodesic focal varieties and the J2-condition
  11. Homogeneity of the sphere-like isoparametric hypersurfaces
  12. Mean curvature of the tubes about the focal varieties

Key Result

Proposition 2.2

If $F$ is an isoparametric function satisfying equations eq:isoparametric_a_b, then the trace $h$ of the shape operator of a regular level set $F^{-1}(c)$ of $F$ with respect to the unit normal vector field $\mathbf N=\frac{\nabla F}{\sqrt{b\circ F}}$ is expressed by

Theorems & Definitions (64)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4: Wang,Ge_Tang
  • Proposition 2.5: Wang
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 54 more