A filtration for isoparametric hypersurfaces in Riemannian manifolds

This paper introduces the notion of k-isoparametric hypersurface in an $(n+1)$-dimensional Riemannian manifold for k=0,1,,n. Many fundamental and interesting results (towards the classification of homogeneous hypersurfaces among other things) are given in complex projective spaces, complex hyperbolic spaces, and even in locally rank one symmetric spaces.