ISOPARAMETRIC HYPERSURFACES IN CP~n WITH CONSTANT PRINCIPAL CURVATURES

This paper proves that the number of distinct principal curvatures of a realisoparametric hypersurface in CP~n with constant principal curvatures can be only 2, 3 or 5.The prehnage of such hypersurface under the Hopf fibration is an isoparametrichypersarface in Swith 2 or 4 distinct principal curvatures. For real isoparametrichypersurfaces in CP~n with 5 distinct constant principal curvatures a local structuretheorem is given.