Principal Distribution Isomorphisms and Almost Hermitian geometry on Isoparametric Hypersurfaces

TL;DR

The paper develops global vector-bundle isomorphisms between principal distributions on OT--FKM isoparametric hypersurfaces in spheres, showing $\mathcal{D}_1 \cong \mathcal{D}_3$ universally and $\mathcal{D}_2 \cong \mathcal{D}_4$ in several multiplicity cases, plus a novel global isomorphism $\mathcal{D}_1 \oplus \mathcal{D}_2 \cong \mathcal{D}_3 \oplus \mathcal{D}_4$ for all odd $m$. These isomorphisms enable natural almost complex-geometric structures, yielding nearly Kähler metrics for certain OT--FKM types and a vanishing $*$-Ricci curvature when an almost Hermitian structure interchanges principal distributions in pairs. A key methodological theme is the use of Clifford-system geometry and Clifford-sphere fibrations to realize explicit, global bundle maps. The results connect the intrinsic isoparametric data with extrinsic almost Hermitian geometry, providing concrete conditions for nearly Kähler structures and $*$-Einstein-type curvature properties, and clarifying obstructions to integrability in higher-multiplicity cases.

Abstract

This paper investigates the isomorphisms between principal distributions $\mathcal{D}_k$ $(k=1,\dots 4)$ on OT--FKM type isoparametric hypersurfaces in spheres. We recover the isomorphism $\mathcal{D}_1 \cong \mathcal{D}_3$ established by Qian--Tang--Yan \cite{Q-T-Y 2}, and further construct the isomorphism $\mathcal{D}_{2}\cong\mathcal{D}_{4}$ in specific cases. More significantly, we provide an explicit construction of a global vector bundle isomorphism $\mathcal{D}_1 \oplus \mathcal{D}_2 \cong \mathcal{D}_3 \oplus \mathcal{D}_4$ for all odd multiplicities $m$. As applications, we employ these isomorphisms to induce nearly Kähler structures on certain OT--FKM hypersurfaces. Finally, we prove that the $*$-Ricci curvature vanishes for any OT--FKM hypersurface admitting an almost Hermitian structure that interchanges principal distributions in pairs.

Theorems & Definitions (25)

• Proposition 1.1
• Proposition 1.1
• Theorem 1.3
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Theorem 2.3