球面中等参超曲面及其焦流形的有限性定理

球面中等参超曲面理论近期获得蓬勃发展,它们在等距意义下的分类至今未完全解决,将证明球面中等参超曲面及其焦流形的微分同胚型有限.     

Singular Riemannian Foliations and Isoparametric Submanifolds

We will explain how polar actions on Riemannian manifolds lead in a natural way to a class of singular Riemannian foliations which turns out to generalize isoparametric foliations of Euclidean spaces and spheres. Both classical and recent results will be explained.     

Space-like Isoparametric Hypersurfaces in Lorentzian Space Forms

The purpose of this paper is to make clear the so-called Nomizu problem, whether it is possible to find examples of space-like isoparametric hypersurfaces in H ₁ ⁿ⁺¹ with more than two distinct principal curvatures. It is proved that a space-like isoparametric hypersurface in H ₁ ⁿ⁺¹ or S ₁ ⁿ⁺¹ can have at most two distinct principal curvatures. The authors present the classification and explicit analytic expressions of such type of isoparametric hypersurfaces. This paper was translated from J. Nanchang Univ. Nat. Sci. Ed., 2004, 28(2): 113–117     

复射影空间的等参子流形

本文给出了复射影空间Ｐ＿ｎ（Ｃ）上的等参映射定义，并证明了等参映射ｆ在Ｈｏｐｆ主丛π：Ｓ￣（２ｎ＋１）→Ｐ＿ｎ（Ｃ）下的水平提升为Ｓ￣（２ｎ＋１）的等参映射。同时，利用对称空间的表示给出了Ｐ＿ｎ（Ｃ）上等参子流形的例子．     

Classifications of Isoparametric Hypersurfaces in Randers Space Forms

In this paper, we give the complete classifications of isoparametric hypersurfaces in Randers space forms. By studying the principal curvatures of anisotropic submanifolds in a Randers space (N, F) with the navigation data (h, W), we find that a Randers space form (N, F, dμ_BH) and the corresponding Riemannian space (N, h) have the same isoparametric hypersurfaces, but in general, their isoparametric functions are different. We give a necessary and sufficient condition for an isoparametric function of (N, h) to be isoparametric on (N, F, dμ_BH), from which we get some examples of isoparametric functions.     

S4中等参超曲面的谱刻画

本文证明了如果Ｓ⁴中的具常平均曲率ｈ的超曲面Ｍ与其具平均曲率ｈ的等参超曲面Ｍ_０（强）等谱，则Ｍ＝Ｍ_０．     

R~6中的Laguerre等参超曲面的分类

若超曲面的Laguerre形式为零且Laguerre第二基本形式的特征值(称为Laguerre主曲率)为常数,则称超曲面为Laguerre等参超曲面.对R~6中的Laguerre等参超曲面进行了研究,得到了分类定理.     

Null Screen Isoparametric Hypersurfaces in Lorentzian Space Forms

In this paper, we develop the notion of screen isoparametric hypersurface for null hypersurfaces of Robertson–Walker spacetimes. Using this formalism we derive Cartan identities for the screen principal curvatures of null screen isoparametric hypersurfaces in Lorentzian space forms and provide a local characterization of such hypersurfaces.     

Isoparametric Hypersurfaces Induced by Navigation in Lorentz Finsler Geometry

Acta Mathematica Sinica, English Series - Using a navigation process with the datum (F, V), in which F is a Finsler metric and the smooth tangent vector field V satisfies F(−V(x)) > 1...     

Invariant Submanifolds of Real Hypersurfaces of Complex Manifolds

Real hypersurfaces of a complex manifold admit a naturally induced almost contact structure F′ from the almost complex structure of the ambient manifold. We prove that for any F′-invariant submanifold M of a geodesic hypersphere in a non-flat complex space form and of a horosphere in a complex hyperbolic space, its second fundamental form h satisfies the condition h(FX,Y ) - h(X, FY) = g(FX, Y )h, X,Y ? T(M), 0 1 h ? T^{^}(M){h(FX,Y ) - h(X, FY) = g(FX, Y )eta, X,Y in T(M), 0 ne eta in {T^perp}(M)}, which has been considered in [2] and [3].     

Notes on the Lightlike Hypersurfaces Along Spacelike Submanifolds

Ukrainian Mathematical Journal - In connection with the method of construction of lightlike hypersurfaces along spacelike submanifolds, we present a relation between the second fundamental form of...     

Compact Ovoids in Quadrangles III: Clifford Algebras and Isoparametric Hypersurfaces

In this third part, we consider those compact quadrangles which arise from isoparametric hypersurfaces of Clifford type and their focal manifolds. Sections 9–11 give a comprehensive introduction to these quadrangles from the incidence-geometric point of view. Section 10 contains also a new (algebraic) proof that these geometries are quadrangles.We determine which of these quadrangles have ovoids or spreads and also whether the normal sphere bundles of the focal manifolds admit sections, or whether they are topologically trivial. We give explicit geometric constructions for spreads, ovoids, and sections.     

Laguerre Isoparametric Hypersurfaces in R~n with Two Distinct Non-zero Principal Curvatures

An umbilical free oriented hypersurfacex:M→Rnwith non-zero principal curvatures is called a Laguerre isoparametric hypersurface if its Laguerre form C=i Ciωi=iρ1(Ei(logρ)(r ri)Ei(r))ωi vanishes and Laguerre shape operator S=ρ1(S 1 rid)has constant eigenvalues.Hereρ=i(r ri)2,r=r1+r2+···+rn 1n 1is the mean curvature radius andSis the shape operator ofx.{Ei}is a local basis for Laguerre metric g=ρ2III with dual basis{ωi}and III is the third fundamental form ofx.In this paper,we classify all Laguerre isoparametric hypersurfaces in Rn(n3)with two distinct non-zero principal curvatures up to Laguerre transformations.     

Classification of the Blaschke Isoparametric Hypersurfaces with Three Distinct Blaschke Eigenvalues

An immersed umbilic-free submanifold in the unit sphere is called Blaschke isoparametric if its Möbius form vanishes identically and all of its Blaschke eigenvalues are constant. Then the classification of Blaschke isoparametric hypersurfaces is natural and interesting in the Möbius geometry of submanifolds. In this paper, we give a classification of the Blaschke isoparametric hypersurfaces with three distinct Blaschke eigenvalues one of which is simple.     

Focal Loci of Algebraic Hypersurfaces: A General Theory

The focal locus is traditionally defined for a differentiable submanifold of R ⁿ. However, since it depends essentially only on the notion of orthogonality, a focal locus can be also associated to an algebraic subvariety of the space , once we have chosen an orthogonal structure on this space. In this paper, we establish somebasic results in the theory of focal loci of algebraichypersurfaces in . Our main results concern the irreducibility of the ramification divisor of the end-point map and the dimension of the singular locus of this divisor, the birationality of the focal map and the degree of the focal locus of an algebraic hypersurface.     

球面上具有四个不同主曲率的Moebius等参超曲面

设x:M→S~(n+1)(n≥5)是(n+1)-维单位球面上不含脐点的超曲面,在S~(n+1)的Moebius变换群下浸入x的四个基本不变量是:Moebius度量g;Moebius第二基本形式B;Moebius形式Φ和Blaschke张量A.本文给出S~(n+1)上具有重数1,1,1,m(m≥2)的四个不同Moebius主曲率的Moebius等参超曲面的分类.     

非紧对称空间中子流形焦点和形算子间的关系

首先详细地讨论了非紧Ｌｉｅ群的度量和Ｃａｒｔａｎ分解,然后由Ｌｉｅ群和对称空间的关系得到了非紧对称空间中的子流形焦点存在的充要条件,同时还给出了焦点重数的计算方法．