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 K\"ahler 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.