A characterization of totally <Emphasis Type="Italic">η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

Three real hypersurfaces some of whose geodesics are mapped to circles with the same curvature in a nonflat complex space form

We characterize all totally η-umbilic hypersurfaces and ruled real hypersurfaces in nonflat complex space forms and certain real hypersurfaces of type (A₂) in complex projective spaces by using the property that some of their geodesics are mapped to circles of the same curvature in these ambient spaces.     

Curvatures properties of Lie hypersurfaces in the complex hyperbolic space

A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In this paper, we study intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures, and sectional curvatures.     

Ruled Real Hypersurfaces in a Nonflat Quaternionic Space Form

In this paper we construct many ruled real hypersurfaces in a nonflat quaternionic space form systematically, and in particular give an example of a homogeneous ruled real hypersurface in a quaternionic hyperbolic space. In the second half of this paper we characterize them by investigating the extrinsic shape of their geodesics. We also characterize curvature-adapted real hypersurfaces in nonflat quaternionic space forms from the same viewpoint.The first author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 14540075), Ministry of Education, Science, Sports and Culture.The second author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 14540080), Ministry of Education, Science, Sports and Culture.     

Quasi—Einstein Hypersurfaces in a Hyperbolic Space

§1.　IntroductionLetRijbethecomponentsofRiccitensorofann-dimensionalRiemannianmanifoldM.IfRij=Agij Bξiξj,　(i,j=1,2,…,n)(1.1)whereξisanunitvectorfield,thenMiscalledaquasi-EinsteinmanifoldanddenotedbyQE(ξ).Ifξisanisotropicvectorfield,thenMiscalledageneralizedquasi-Einsteinmanifold.Intheequality(1.1),AandBarescalarfunctions.WeknowQE(ξ)manifoldisEinsteinwhenB≡0.Especially,if〈ξ,ξ〉=e=±1,thenQE(ξ)iscalledanormalquasi-Einsteinmani-fold.Itiseasytoknowfrom[1]and[2]:Rij=R-Tn-1…     

The classification of golden shaped hypersurfaces in Lorentz space forms

In this paper, we will study the golden shaped hypersurfaces in Lorentz space forms. Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the golden shaped hypersurfaces in Minkowski space, de Sitter space and anti-de Sitter space, respectively.     

四元双曲空间中的实超曲面

设M是四元双曲空间中的实超曲面,若M是Weingarten形状算子A相对于三个特定方向平行,则M是一个管状超曲面。     

A characterization of a certain real hypersurface of type (A<Subscript>2</Subscript>) in a complex projective space

In the class of real hypersurfaces M ^2n?1 isometrically immersed into a nonflat complex space form \(\widetilde {{M_n}}\left( c \right)\) of constant holomorphic sectional curvature c (≠ 0) which is either a complex projective space ?P ⁿ (c) or a complex hyperbolic space ?H ⁿ (c) according as c > 0 or c < 0, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in \(\widetilde {{M_n}}\left( c \right)\), we consider a certain real hypersurface of type (A₂) in ?P ⁿ (c) and give a geometric characterization of this Hopf manifold.     

复射影空间中具有常数量曲率的实超曲面

将球面上常数量曲率超曲面推广到复射影空间中,得到此类实超曲面的某些积分不等式.     

Finiteness results for equifocal hypersurfaces in compact symmetric spaces

We prove that there are only finitely many diffeomorphism types of curvature-adapted equifocal hypersurfaces in a simply connected compact symmetric space. Moreover, if the symmetric space is of rank one, the result can be strengthened by dropping the condition curvature-adapted.     

Circular trajectories on real hypersurfaces in a nonflat complex space form

In this paper we study which trajectories for Sasakian magnetic fields are circles on certain standard real hypersurfaces which are called hypersurfaces of type A in a nonflat complex space form. We also give a characterization of these real hypersurfaces by such a circular property of trajectories for Sasakian magnetic fields.     

Curvature of Hopf Hypersurfaces in a Complex Space Form

We study curvature of Hopf hypersurfaces in a complex projective space or hyperbolic space. In particular, we prove that there are no real hypersurfaces in a non-flat complex space form whose Reeb-sectional curvature vanishes.     

On the Riemannian curvature tensor of a real hypersurface in a complex projective space

We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections and classify them when the covariant derivatives associated with both connections, either in the direction of the structure vector field or any direction of the maximal holomorphic distribution, coincide when applying to the Riemannian curvature tensor of the real hypersurface.     

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.     

Spacelike Möbius Hypersurfaces in Four Dimensional Lorentzian Space Form

In this paper, we first set up an alternative fundamental theory of Möbius geometry for any umbilic-free spacelike hypersurfaces in four dimensional Lorentzian space form, and prove the hypersurfaces can be determined completely by a system consisting of a function W and a tangent frame {E_i}. Then we give a complete classification for spacelike Möbius homogeneous hypersurfaces in four dimensional Lorentzian space form. They are either Möbius equivalent to spacelike Dupin hypersurfaces or to some cylinders constructed from logarithmic curves and hyperbolic logarithmic spirals. Some of them have parallel para-Blaschke tensors with non-vanishing Möbius form.     

Lokale Kennzeichnungen der Minimalhyperregelflächen

Generalized ruled hypersurfaces are generated by a one-parameter family of (n–2)-dimensional linear subspaces of the n-dimensional Euclidean space E_n. In this paper we give local characterizations of generalized ruled hypersurfaces with an everywhere vanishing mean curvature.     

Ricci Curvature of Real Hypersurfaces in Non-flat Complex Space Forms

We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.     

Die Minimalhypekregelflächen

In this paper we investigate (k+1)-dimensional generalized ruled surfaces generated by a one-parameter family of k-dimensional linear subspaces of the n-dimensional Euclidean space E_n. All generalized ruled hypersurfaces (i. e. (n–1)-dimensional generalized ruled surfaces) with an everywhere vanishing mean curvature are listed. In other words, a complete characterization is given of all minimal ruled hypersurfaces. Moreover the principal curvatures of these surfaces are determined.     

Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by J be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field N is said to be Hopf if the tangent vector field JN is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the Hopf curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.