On Hopf hypersurfaces of the complex quadric with constant principal curvatures

TL;DR

The article resolves the classification of Hopf real hypersurfaces in the complex quadric $Q^m$ with constant principal curvatures when the number of distinct curvatures is at most five, showing that such hypersurfaces are open parts of tubes over totally geodesic $Q^{m-1}$ or $\mathbb{C}P^k\hookrightarrow Q^{2k}$, with explicit results for $m=3,4,5$. It introduces a new family with $\mathfrak{A}$-isotropic normals, derives two Cartan-type formulas tailored to $Q^m$, and proves that parallel hypersurfaces remain isoparametric with austere focal submanifolds. The analysis leverages Jacobi-field methods to compute parallel/focal curvatures and uses the interplay between the shape operator, complex structure $J$, and almost product structure $A$. As a consequence, the paper establishes nonexistence of five-distinct-curvature examples, describes the six-curvature phenomenon in $Q^6$, and yields sharp classifications in low dimensions, thereby advancing the understanding of real hypersurfaces in complex quadrics. The findings have significance for the geometry of Hermitian symmetric spaces and the theory of isoparametric and austere submanifolds in this setting.

Abstract

In this paper, we classify the Hopf hypersurfaces of the complex quadric $Q^m=SO_{m+2}/(SO_2SO_m)$ ($m\geq3$) with at most five distinct constant principal curvatures. We also classify the Hopf hypersurfaces of $Q^m$ ($m=3,4,5$) with constant principal curvatures. All these real hypersurfaces are open parts of homogeneous examples.

Theorems & Definitions (60)

• Theorem 1.1
• Remark 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.2
• Lemma 2.1: cf. RSm
• Definition 2.2: BSBS-2022Loo11
• Remark 2.1