Real hypersurfaces in a complex space form with a condition on the structure Jacobi operator

In this paper we classify the real hypersurfaces in a non-flat complex space form with its structure Jacobi operator R _ξ satisfying (? _X R _ξ )ξ = 0, for all vector fields X in the maximal holomorphic distribution D. With this result, we prove the non-existence of real hypersurfaces with D-parallel as well as D-recurrent structure Jacobi operator in complex projective and hyperbolic spaces. We can also prove the non-existence of real hypersurfaces with recurrent structure Jacobi operator in a non-flat complex space form as a corollary.     

Geometry of Holomorphic Distributions of Real Hypersurfaces in a Complex Projective Space

We characterize homogeneous real hypersurfaces M's of type (A ₁), (A ₂) and (B) of a complex projective space in the class of real hypersurfaces by studying the holomorphic distribution T ⁰ M of M.     

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.     

Differential geometry of real submanifolds in a Kähler manifold

LetN be a real submanifold in a complex manifoldM. If the maximal complex subspaces of the tangent spaces ofM contained in the tangent spaces ofN are of constant dimension and they define a differentiable distribution, thenN is called a generic submanifold. The class of generic submanifold includes all real hypersurfaces, complex submanifolds, totally real submanifolds andCR-submanifolds. In this paper we initiate a study of generic submanifolds in a Kähler manifold from differential geometric point of view. Some fundamental results in this respect will be obtained.     

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].     

Congruence of hypersurfaces inS 6 and inC P n

The invariants needed to decide when a pair of hypersurfaces ofS ⁶ orCP ⁿ are respectivelyG ₂-congruent or holomorpic congruent are determined and this result is used to characterize the hypersurfaces of these spaces whose Hopf vector fields are also Killing fields.     

Recurrent real hypersurfaces in complex two-plane Grassmannians

Summary We introduce the notion of recurrent shape operator for a real hypersurface M in the complex two-plane Grassmannians G₂(C^m+2) and give a non-existence property of real hypersurfaces in G₂(C^m+2) with the recurrent shape operator.     

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.     

Real hypersurfaces of a complex projective space

In this paper, we classify real hypersurfaces in the complex projective space C P^\fracn+12C P^{\frac{n+1}{2}} whose structure vector field is a φ-analytic vector field (a notion similar to analytic vector fields on complex manifolds). We also define Jacobi-type vector fields on a Riemannian manifold and classify real hypersurfaces whose structure vector field is a Jacobi-type vector field.     

Geodesics on real hypersurfaces of type A 2 in a complex space form

In this paper, we study geodesics with null structure torsions on real hypersurfaces of type A ₂ in a complex space form. These geodesics give a nice family of helices of order 3 generated by Killing vector fields on the ambient complex space form. Author’s address: Toshiaki Adachi, Department of Mathematics, Nagoya Institute of Technology, Nagoya 466-8555, Japan     

Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

 We classify all real hypersurfaces with isometric Reeb flow in the complex Grassmann manifold G ₂ (ℂ ^m+2 ) of all 2-dimensional linear subspaces in ℂ ^m+2 , m ≥ 3.     

On a family of lie algebras related to homogeneous surfaces

Real affine homogeneous hypersurfaces of general position in three-dimensional complex space ?³ are studied. The general position is defined in terms of the Taylor coefficients of the surface equation and implies, first of all, that the isotropy groups of the homogeneous manifolds under consideration are discrete. It is this case that has remained unstudied after the author’s works on the holomorphic (in particular, affine) homogeneity of real hypersurfaces in three-dimensional complex manifolds. The actions of affine subgroups G ? Aff(3, ?) in the complex tangent space T ^? _p M of a homogeneous surface are considered. The situation with homogeneity can be described in terms of the dimensions of the corresponding Lie algebras. The main result of the paper eliminates “almost trivial” actions of the groups G on the spaces T _p ^? M for affine homogeneous strictly pseudoconvex surfaces of general position in ?³ that are different from quadrics.     

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.     

Viro theorem and topology of real and complex combinatorial hypersurfaces

We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension 2 inℂP ⁿ and are topologically “glued” out of algebraic hypersurfaces in (ℂ^*) ⁿ . Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces. A part of the present work was done during the stay of the second author at the Fields Institute, Toronto, and at the NSF Science and Technology Research Center for the Computation and Visualization of Geometric Structures, funded by NSF/DMS89-20161. The work was completed during the stay of both authors at Max-Planck-Institu für Mathematik. The authors thank these funds and institutions for hospitality and financial support.     

Low-type submanifolds of real space forms via the immersions by projectors

We undertake a comprehensive study of submanifolds of low Chen-type (1, 2, or 3) in non-flat real space forms, immersed into a suitable (pseudo) Euclidean space of symmetric matrices by projection operators. Some previous results for submanifolds of the unit sphere (obtained in [A. Ros, Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z. 187 (1984) 393–404; M. Barros, B.Y. Chen, Spherical submanifolds which are of 2-type via the second standard immersion of the sphere, Nagoya Math. J. 108 (1987) 77–91; I. Dimitrić, Spherical hypersurfaces with low type quadric representation, Tokyo J. Math. 13 (1990) 469–492; J.T. Lu, Hypersurfaces of a sphere with 3-type quadric representation, Kodai Math. J. 17 (1994) 290–298]) are generalized and extended to real projective and hyperbolic spaces as well as to the sphere. In particular, we give a characterization of 2-type submanifolds of these space forms with parallel mean curvature vector. We classify 2-type hypersurfaces in these spaces and give two sets of necessary conditions for a minimal hypersurface to be of 3-type and for a hypersurface with constant mean curvature to be mass-symmetric and of 3-type. These conditions are then used to classify such hypersurfaces of dimension n5. For example, the complete minimal hypersurfaces of the unit sphere Sⁿ⁺¹ which are of 3-type via the immersion by projectors are exactly the 3-dimensional Cartan minimal hypersurface and the Clifford minimal hypersurfaces M_k,n−k for n≠2k. An interesting characterization of horospheres in is also obtained.     

Real hypersurfaces in complex two-plane grassmannians with parallel structure Jacobi operator

We give some non-existence theorems for Hopf real hypersurfaces in complex two-plane Grassmannians G ₂(? ^m+2) with parallel structure Jacobi operator R _ξ.     

Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

 We classify all real hypersurfaces with isometric Reeb flow in the complex Grassmann manifold G ₂ (ℂ ^m+2 ) of all 2-dimensional linear subspaces in ℂ ^m+2 , m ≥ 3. The second author was supported by Korea Research Foundation. KRF-2001-015-DP0034, Korea. Received April 26, 2001; in revised form December 17, 2001     

Anisotropic isoparametric hypersurfaces in Euclidean spaces

In this paper, by Nomizu’s method and some technical treatment of the asymmetry of the F-Weingarten operator, we obtain a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is a generalization of the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local anisotropic isoparametric surface constructed by B. Palmer, we find that in general anisotropic isoparametric hypersurfaces have both local and global aspects as in the theory of proper Dupin hypersurfaces, which differs from classical isoparametric hypersurfaces.     

Minimal and Harmonic Unit Vector Fields in and Its Dual Space

 The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results hold for the dual space .     

On the ricci tensor of a real hypersurface of quaternionic hyperbolic space

Summary We prove the non-existence of Einstein real hypersurfaces of quaternionic hyperbolic space. This article was processed by the author using the IAT_EX style filecljour1 from Springer-Verlag.