Submanifolds in complex projective and hyperbolic planes

Abstract

In this PhD thesis we study submanifolds in complex projective and hyperbolic spaces. More specifically, we classify isoparametric and Terng-isoparametric submanifolds. The former correspond to principal orbits of polar actions, whereas the latter are homogeneous but not necessarily arising from polar actions. We also study real hypersurfaces with two distinct principal curvatures, show that there are non-Hopf inhomogeneous examples, and characterize them. Using the method of equivariant geometry, we investigate strongly 2-Hopf hypersurfaces and give some applications for Levi-flat and constant mean curvature hypersurfaces. Finally, we classify austere hypersurfaces such that the number of nontrivial projections of the Hopf vector field onto the principal curvature spaces is less or equal than two; all the examples are ruled in this case.