Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

We classify real hypersurfaces with constant principal curvatures in the complex hyperbolic plane. It follows from this classification that all of them are open parts of homogeneous ones.

     

On hypersurfaces with two distinct principal curvatures in space forms

We investigate the immersed hypersurfaces in space forms ℕ ^n + 1(c), n ≥ 4 with two distinct non-simple principal curvatures without the assumption that the (high order) mean curvature is constant. We prove that any immersed hypersurface in space forms with two distinct non-simple principal curvatures is locally conformal to the Riemannian product of two constant curved manifolds. We also obtain some characterizations for the Clifford hypersurfaces in terms of the trace free part of the second fundamental form.     

Real hypersurfaces with constant totally real bisectional curvature in complex space forms

In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space form M _m (c), c ≠ 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].     

On spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

We investigate the spacelike hypersurfaces in Lorentzian space forms (n?4) with two distinct non-simple principal curvatures without the assumption that the (high order) mean curvature is constant. We prove that any spacelike hypersurface in Lorentzian space forms with two distinct non-simple principal curvatures is locally conformal to the Riemannian product of two constant curved manifolds. We also obtain some characterizations for hyperbolic cylinders in Lorentzian space forms in terms of the trace free part of the second fundamental form.     

Real hypersurfaces in complex space forms concerned with the local symmetry

This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface M in complex space form M _m(4ε). In the second, we give a complete classification of real hypersurfaces in M _m(4ε) which satisfy the above geometric facts. The second author was supported by DGICYT research project BFM 2001-2871-C04-01 and the first and the third authors were supported by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006.     

On the complete linear Weingarten spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

We deal with complete linear Weingarten spacelike hypersurfaces immersed in a Lorentzian space form, having two distinct principal curvatures. In this setting, we show that such a spacelike hypersurface must be isometric to a certain isoparametric hypersurface of the ambient space, under suitable restrictions on the values of the mean curvature and of the norm of the traceless part of its second fundamental form. Our approach is based on the use of a Simons type formula related to an appropriated Cheng–Yau modified operator jointly with some generalized maximum principles.     

Dupin hypersurfaces with three principal curvatures

Using the method of moving frames, we prove that any locally irreducible Dupin hypersurface in S ⁿ with three distinct principal curvatures is equivalent by Lie sphere transformation to an isoparametric hypersurface in S ⁿ . Oblatum VI-1995 & 28-IV-1997     

Conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in conformal space

The conformal geometry of regular hypersurfaces in the conformal space is studied.We classify all the conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in the conformal space up to conformal equivalence.     

Compact Hopf hypersurfaces of constant mean curvature in complex space forms

We prove that every connected compact Hopf hypersurface of a complex space form, contained in a geodesic ball of radius strictly smaller than the injectivity radius of, having constant mean curvature and with if if < 0 is a geodesic sphere of.Work partially supported by DGICYT Grant No. PB91-0324.     

CMC hypersurfaces with canonical principal direction in space forms

A hypersurface of the space form has a canonical principal direction (CPD) relative to the closed and conformal vector field Z of if the projection of Z to M is a principal direction of M . We show that CPD hypersurfaces with constant mean curvature are foliated by isoparametric hypersurfaces. In particular, we show that a CPD surface with constant mean curvature of space form is invariant by the flow of a Killing vector field whose action is polar on . As consequence we show that a compact CPD minimal surface of the sphere is a Clifford torus. Finally, we consider the case when a CPD Euclidean hypersurface has zero Gauss–Kronecker curvature.     

Real hypersurfaces with constant totally real sectional curvature in a complex space form

We characterize real hypersurfaces with constant holomorphic sectional curvature of a non flat complex space form as the ones which have constant totally real sectional curvature.     

Complete rotation hypersurfaces with Hk constant in space forms

In this paper we classify all complete rotation hypersurfaces withH _k constant in ⁿ⁺¹ andH ⁿ⁺¹, is the normalizedk-th symmetric function of the principal curvatures. Partial results are also given forH ⁿ⁺¹.Partially supported by DGAPA-UNAM, México, CONACYT, México, under Project 1068P, and CNPp, Brazil.