Topology and Curvature of Isoparametric Families in Spheres

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 Wang (J Differ Geom 27:55-66, 1988). 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.