On the Classification of Isoparametric Hypersurfaces with Constant Principal Curvatures in Compact 3-Manifolds

Abstract

Establishing detailed relationships between transnormal systems of different types and their behaviors under covering maps, this paper presents a classification of transnormal systems on compact 3-manifolds in the sense of equivalence. For CPC transnormal systems, we show that the ambient manifolds must be locally isometric to one of six standard geometries up to equivalence. We also find some equivalence classes containing no CPC transnormal system, highlighting a critical distinction between isoparametric foliations and CPC transnormal systems, which has not been previously addressed in the literature.