Totally umbilic null hypersurfaces in generalized Robertson–Walker spaces

We show that there is a correspondence between totally umbilic null hypersurfaces in generalized Robertson–Walker spaces and twisted decompositions of the fibre. This allows us to prove that nullcones are the unique totally umbilic null hypersurfaces in the closed Friedmann Cosmological model. We also apply this kind of ideas to static spaces, in particular to Reissner–Nordström and Schwarzschild exterior spacetimes.     

Ruled surfaces with constant terms in the relative Theorema Egregium

Two complete classifications are given: (a) relative surfaces with isoparametric relative shape operator and projectively flat relative induced connection and (b) relative Tchebychev surfaces with constant terms in the relative Theorema Egregium. Both classifications share the occurrence of certain classes of ruled surfaces satisfying these strong conditions with respect to a relative geometry.     

Minkowski空间中带超平面边界的紧致类空超曲面

In this paper, we study the relations between a compact spacelike hypersurface with hy-perplanar boundary in the (n 1)- dimensional Minkowski space-time L^n 1 being totally umbilicaland its hyperplanar boundary in a fixed hyperplane π of L^(n 1) being totally umbilical under certainconditions. We give the sufficient conditions for such hypersurface and its hyperplanar boundary tobe totally umbilical in their respective ambients.     

Real Hypersurfaces in Quaternionic Space Forms Satyisfying Axioms of Planes

A Riemannian manifold satisfies the axiom of 2-planes if at each point, there are suficiently many totally geodesic surfaces passing through that point. Real hypersurfaces in quaternionic space forms admit nice families of tangent planes, namely, totally real, half-quaternionic and quaternionic. Several definitions of axiom of planes arise naturally when we consider such families of tangent planes. We are able to classify real hypersurfaces in quaternionic space forms satisfying these definitions.     

<Emphasis Type="Italic">p</Emphasis>-Laplace Operator and Diameter of Manifolds

Let be a compact Riemannian manifold without boundary. In this paper, we consider the first nonzero eigenvalue of the p-Laplacian and we prove that the limit of when is 2/d(M), where d(M) is the diameter of M. Moreover, if is an oriented compact hypersurface of the Euclidean space or , we prove an upper bound of in terms of the largest principal curvature κ over M. As applications of these results, we obtain optimal lower bounds of d(M) in terms of the curvature. In particular, we prove that if M is a hypersurface of then: . Mathematics Subject Classifications (2000): 53A07, 53C21.     

Möbius Homogeneous Hypersurfaces with One Simple Principal Curvature in Sn+1

Let Möb(Sⁿ⁺¹) denote the Möbius transformation group of Sⁿ⁺¹. A hypersurface f:Mⁿ → Sⁿ⁺¹ is called a Möbius homogeneous hypersurface, if there exists a subgroup G ? Möb(Sⁿ⁺¹) such that the orbit G(p)={φ(p)|φ ∈ G}=f(Mⁿ), p ∈ f(Mⁿ). In this paper, we classify the Möbius homogeneous hypersurfaces in Sⁿ⁺¹ with at most one simple principal curvature up to a Möbius transformation.     

An Intrinsic Rigidity Theorem for Closed Minimal Hypersurfaces in S5 with Constant Nonnegative Scalar Curvature

Let M⁴ be a closed minimal hypersurface in \(\mathbb{S}^5\) with constant nonnegative scalar curvature. Denote by f₃ the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f₃ and g are constant, then M⁴ is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form of M⁴. This result provides another piece of supporting evidence to the Chern conjecture.     

Transversely holomorphic structures with many invariant hypersurfaces

In this paper we show that a transversely holomorphic foliation in a compact manifold $M$ having an infinite number of invariant hypersurfaces admits a basic transversely meromorphic first integral $f:M?\overline{\mathbb{C}}$.     

Classification of conformally flat isoparametric submanifolds of Euclidean space

In this note we provide a direct proof of the complete classification of conformally flat isoparametric submanifolds of Euclidean space.     

Lorentz空间型中正常2-调和超曲面的分类

本文对Lorentz空间型中的正常2-调和超曲面进行了完全分类,它的形状算子的极小多项式的阶数至多是2.