Topology and curvature of isoparametric families in spheres

TL;DR

This paper studies the topology and curvature properties of isoparametric families in spheres, extending previous results on their topological classifications and identifying conditions for non-negative sectional and positive Ricci curvatures.

Contribution

It advances understanding of the topology and curvature of isoparametric families, extending prior work and providing new curvature criteria.

Findings

Extended topological classifications of isoparametric families.

Identified conditions for non-negative sectional curvatures.

Determined when these families have positive Ricci curvatures.

Abstract

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik-Schnirelmann category. This part extends substantially the results of Q.M.Wang in \cite{Wa88}. The second part is concerned with their curvatures, more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.