On spacelike hypersurfaces with constant scalar curvature in the de Sitter space

We classify spacelike hypersurfaces of the de Sitter space with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we prove several rigidity theorems for such hypersurfaces.     

On spacelike hypersurfaces with constant scalar curvature in the anti-de Sitter space

In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we also obtain several rigidity theorems for such hypersurfaces.     

Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds

Our purpose in this paper is to study the rigidity of complete linear Weingarten hypersurfaces immersed in a locally symmetric manifold obeying some standard curvature conditions (in particular, in a Riemannian space with constant sectional curvature). Under appropriated constrains on the scalar curvature function, we prove that such a hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of them being simple. Furthermore, we also deal with the parabolicity of these hypersurfaces with respect to a suitable Cheng–Yau modified operator.     

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.     

Prescribed Mean Curvature Hypersurfaces in Hn+1(-1) with Convex Planar Boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hⁿ⁺¹(-1). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.     

On Dupin Hypersurfaces in R5 Parametrized by Lines of Curvature

We provide a complete local characterization of Dupin hypersurfaces in R⁵, with four distinct principal curvatures, parametrized by lines of curvature. Such hypersurfaces are given in terms of the principal curvatures and vector valued functions that describe plane curves. We include explicit examples of such Dupin hypersurfaces which are irreducible and have nonconstant Lie curvature.     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

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.     

A finiteness theorem for isoparametric hypersurfaces

In this note we prove that for eachn there are only finitely many diffeomorphism classes of compact isoparametric hypersurfaces ofS ⁿ⁺¹ with four distinct principal curvatures.     

Complete spacelike hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we investigate the complete spacelike hypersurfaces with constant mean curvature and two distinct principal curvatures in an anti-de Sitter space. We give a characterization of hyperbolic cylinder and prove the conjecture in a paper by L. F. Cao and G. X. Wei [J. Math. Anal. Appl., 2007, 329(1): 408–414].     

The decomposition of curvature netted hypersurfaces

We define the concept of a curvature netted hypersurface and investigate in what case the hypersurface is covered by a twisted product of spheres (or topological product of spheres). All hypersurfaces in a space form such that the number of mutually distinct principal curvatures is constant (i.e. each principal curvature has constant multiplicity) are curvature netted hypersurfaces. Also, we state some inductive constructions of the hypersurfaces, where we use the discussion related to the tube.     

On Biconservative Lorentz Hypersurface with Non-diagonalizable Shape Operator

In this paper, we obtain some properties of biconservative Lorentz hypersurface \(M_{1}^{n}\) in \(E_{1}^{n+1}\) having shape operator with complex eigenvalues. We prove that every biconservative Lorentz hypersurface \(M_{1}^{n}\) in \(E_{1}^{n+1}\) whose shape operator has complex eigenvalues with at most five distinct principal curvatures has constant mean curvature. In addition, we investigate such a type of hypersurface with constant length of second fundamental form having six distinct principal curvatures.     

Spacelike Hypersurfaces with Constant Scalar Curvature in the Lorentz–Minkowski Space

We show that the only compact spacelike hypersurfaces in the Lorentz–Minkowski space Lⁿ⁺¹ having nonzero constant scalar curvature and spherical boundary are the hyperbolic caps (with negative constant scalar curvature). One key ingredient in our proof will be an integral formula for the n-dimensional volume enclosed by the boundary of a compact spacelike hypersurface, in the case where the boundary is contained in a hyperplane of Lⁿ⁺¹. As a direct application of that integral formula we also derive an interesting result for the volume of spacelike hypersurfaces.     

On hypersurfaces with two distinct principal curvatures in a unit sphere

We investigate the immersed hypersurfaces in a unit sphere . By using Otsuki's idea, we obtain the local and global classification results for immersed hypersurfaces in of constant m-th mean curvature and two distinct principal curvatures of multiplicities n−1,1 (in the local version, we assume that the principal curvatures are non-zero when m2). As the result, we prove that any local hypersurface in of constant mean curvature and two distinct principal curvatures is an open part of a complete hypersurface of the same curvature properties. The corresponding result does not hold for m-th mean curvature when m2.     

Real hypersurfaces in quaternionic projective space

Summary This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. We obtain three types of such real hypersurfaces. Two of them are known. Third type is new and in its study the first example of proper quaternion CR-submanifold appears. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. Finally we study the Ricci tensor of a real hypersurface of quaternionic projective space and classify pseudo-Einstein, almost-Einstein and Einstein real hypersurfaces.     

An application of maximum principle to space-like hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space ${H^{n+1}_1(-1)}$ . we prove that if a complete space-like hypersurface with constant mean curvature ${x:\mathbf M\rightarrow H^{n+1}_1(-1) }$ has two distinct principal curvatures ??, ??, and inf|?? ? ??|?>?0, then x is the standard embedding ${ H^{m} (-\frac{1}{r^2})\times H^{n-m} ( -\frac{1}{1 - r^2} )}$ in anti-de Sitter space ${ H^{n+1}_1 (-1) }$ .     

Conformally Flat Hypersurfaces of E4

We consider 3-dimensional conformally flat hypersurfaces of E ⁴ with 2 different principal curvatures such that the coordinate directions are principal directions. We describe explicitly those which allow an immersion with constant mean curvature. They are shown to be in close correspondence with solutions of the nonlinear integrable sine-Gordon and sinh-Gordon equations. Conversely, this provides a geometrical characterization for this particular class of conformally flat hypersurfaces of E ⁴.     

Real hypersurfaces with constant principal curvatures in complex space forms

We present the motivation and current state of the classification problem of real hypersurfaces with constant principal curvatures in complex space forms. In particular, we explain the classification result of real hypersurfaces with constant principal curvatures in nonflat complex space forms and whose Hopf vector field has nontrivial projection onto two eigenspaces of the shape operator. This constitutes the following natural step after Kimura and Berndt?s classifications of Hopf real hypersurfaces with constant principal curvatures in complex space forms.     

Dupin Hypersurfaces with Four Principal Curvatures

Using the method of moving frames, we prove that any irreducible Dupin hypersurface in S ⁵ with four distinct principal curvatures and constant Lie curvature is equivalent by Lie sphere transformation to an isoparametric hypersurface in S ⁵.     

Complete Linear Weingarten Spacelike Hypersurfaces Immersed in a Locally Symmetric Lorentz Space

In this paper, we establish a characterization theorem concerning the complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, whose sectional curvature is supposed to obey certain appropriated conditions. Under a suitable restriction on the length of the second fundamental form, we prove that a such spacelike hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple.