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Curvature homogeneous hypersurfaces in space forms

Robert Bryant, Luis Florit, Wolfgang Ziller

TL;DR

This work classifies curvature homogeneous hypersurfaces in space forms $S^4$ and $H^4$, showing that in higher dimensions the landscape is largely governed by the Ferus–Karcher–Münzner (FKM) isoparametric examples and Tsukada’s isolated $H^5$ example. The authors implement the Gauss parametrization, recast the problem in terms of polar surfaces $g:V^2\to S_c^{4}$, and reduce the classification to constant-determinant endomorphisms of the polar shape operators, thereby translating a 4D rank-two problem into a 2D surface problem. They derive and analyze the structure equations, perform a compatibility analysis to identify viable parameter regimes, and establish existence results, including a one-parameter family of polar surfaces and a rotationally symmetric subfamily with explicit parametrizations. The paper culminates with a detailed treatment of the unique $H^5$ case (Tsukada’s theorem) via the Veronese construction, showing that the complete curvature homogeneous hypersurface in $\mathbb H^5$ arises from the unit normal bundle over a Veronese embedding, and relates these findings to the known $S^4$ cases. Overall, the work provides a unified, constructive description of curvature homogeneous rank-two hypersurfaces in low-dimensional space forms and clarifies the rigidity and symmetry features that distinguish the exceptional Tsukada and Veronese-type examples.

Abstract

We classify curvature homogeneous hypersurfaces in S^4 and H^4. In higher dimesnsion one only has the FKM examples and an isolate one by Tsukada of a hypersurface in H^5. Besides some simple examples, we show that there exists an isolated hypersurface with a circle of symmetries and and a one parameter family admitting no continuous symmetries. Outside the set of minimal points, which only exists in the case of S^4, every example is locally and up to covers of this form.

Curvature homogeneous hypersurfaces in space forms

TL;DR

This work classifies curvature homogeneous hypersurfaces in space forms and , showing that in higher dimensions the landscape is largely governed by the Ferus–Karcher–Münzner (FKM) isoparametric examples and Tsukada’s isolated example. The authors implement the Gauss parametrization, recast the problem in terms of polar surfaces , and reduce the classification to constant-determinant endomorphisms of the polar shape operators, thereby translating a 4D rank-two problem into a 2D surface problem. They derive and analyze the structure equations, perform a compatibility analysis to identify viable parameter regimes, and establish existence results, including a one-parameter family of polar surfaces and a rotationally symmetric subfamily with explicit parametrizations. The paper culminates with a detailed treatment of the unique case (Tsukada’s theorem) via the Veronese construction, showing that the complete curvature homogeneous hypersurface in arises from the unit normal bundle over a Veronese embedding, and relates these findings to the known cases. Overall, the work provides a unified, constructive description of curvature homogeneous rank-two hypersurfaces in low-dimensional space forms and clarifies the rigidity and symmetry features that distinguish the exceptional Tsukada and Veronese-type examples.

Abstract

We classify curvature homogeneous hypersurfaces in S^4 and H^4. In higher dimesnsion one only has the FKM examples and an isolate one by Tsukada of a hypersurface in H^5. Besides some simple examples, we show that there exists an isolated hypersurface with a circle of symmetries and and a one parameter family admitting no continuous symmetries. Outside the set of minimal points, which only exists in the case of S^4, every example is locally and up to covers of this form.
Paper Structure (9 sections, 10 theorems, 62 equations, 2 figures)

This paper contains 9 sections, 10 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Gauss parametrization for rank two hypersurfaces
  4. Reduction of the structure equations
  5. Compatibility analysis
  6. The case c=-1 and h=1
  7. Existence of solutions
  8. The rotationally symmetric case
  9. The unique example in H5

Key Result

Theorem 1

. Let ${\cal M}$ be the set of immersed rank two hypersurfaces in $\mathbb Q_c^4$, $c=\pm 1$, whose induced metric has constant scalar curvature. Then ${\cal M}$ contains $f_c$ as the only complete example, an isolated hypersurface $\hat{f}_c$ with a circle of symmetries, and a one parameter family

Figures (2)

  • Figure 1: Leaves of the foliations $D$ for $c=1$ and $c=-1$
  • Figure 2: The curve $\beta$, in blue for $c=1$ and green for $c=-1$

Theorems & Definitions (17)

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