Hipersuperficies con curvaturas principais constantes nos espazos proxectivo e hiperbólico complexos

A homogeneous submanifold  of a Riemannian  manifold is an orbit  of the action  of a closed subgroup  of the isometry group of the ambient manifold. Of particular interest are homogeneous hypersurfaces,  which arise as principal  orbits of cohomogeneity one actions. An important problem in submanifold  geometry is to classify homogeneous submanifolds of a given Riemannian  manifold and to characterize  them  in terms  of geometric  data. A hypersurface  has constant principal  curvatures if the eigenvalues of its shape operator are constant. Obviously, homogeneous hypersurfaces  have constant principal  curvatures. It is still an outstanding open problem to determine under  which circumstances hypersurfaces with constant  principal  curvatures  of a Riemannian  manifold are open parts of homogeneous ones.