Principal Distribution Isomorphisms and Almost Hermitian geometry on Isoparametric Hypersurfaces

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 $\ast$-Ricci curvature vanishes for any OT--FKM hypersurface admitting an almost Hermitian structure that interchanges principal distributions in pairs.

15 pages