Hopf Hypersurfaces Research Articles

The Ricci tensor and structure Jacobi operator of real hypersurfaces in a complex projective space

Let M be a real hypersurface with almost contact metric structure \({(\phi, \xi, \eta, g)}\) in a complex projective space \({P_{n}\mathbb{C}}\) . A Real hypersurface M is said to be a Hopf hypersurface if ξ is principal. In this paper we investigate real hypersurfaces of \({P_{n}\mathbb{C}}\) whose Ricci tensors S satisfy \({\nabla_{\phi\nabla_{\xi}\xi}S = 0}\) . Under some further conditions we characterize Hopf hypersurfaces of \({P_{n}\mathbb{C}}\) .

Real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator is of Codazzi type

We give a non-existence theorem for Hopf hypersurfaces in complex two-plane Grassmannians G 2(ℂm+2) whose structure Jacobi operator R ξ is of Codazzi type.

Hopf hypersurfaces of small Hopf principal curvature in $${\mathbb{C}{\rm H}^2}$$

Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space \({\mathbb{C}{\rm H}^2}\) with any specified value of the Hopf principal curvature α less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces in terms of Weierstrass-type data, and also obtain a classification of pseudo-Einstein hypersurfaces in \({\mathbb{C}{\rm H}^2}\).

Structure Jacobi Operator of Real Hypersurfaces with Constant Scalar Curvature in a Nonflat Complex Space Form

Let $M$ be a real hypersurface with almost contact metric structure $(\phi ,\xi ,\eta ,g)$ in a nonflat complex space form $M_{n}(c)$. We denote by $S$ be the Ricci tensor of $M$. In the present paper we investigate real hypersurfaces with constant scalar curvature of $M_{n}(c)$ whose structure Jacobi operator $R_{\xi}$ commute with both $\phi$ and $S$. We characterize Hopf hypersurfaces of $M_{n}(c)$.

Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

In this paper, we have considered new commuting conditions, that is, (R ξ ϕ)S = S(R ξ ϕ) (resp. \((\bar{R}_{N}\phi )S = S(\bar{R}_{N}\phi )\)) between the Jacobi operators R ξ (resp. \(\bar{R}_{N}\)), the structure tensor field ϕ and the Ricci tensor S for real hypersurfaces M in \(G_{2}(\mathbb{C}^{m+2})\). With such a condition we give a complete classification of Hopf hypersurfaces M in \(G_{2}(\mathbb{C}^{m+2})\).

Geodesic spheres in a nonflat complex space form and their integral curves of characteristic vector fields

For Hopf hypersurfaces in a nonflat complex space form $M^n(c; \Bbb{C})$, integral curves of their characteristic vector fields are ''nice'' curves in the sense that their extrinsic shapes in $M^n(c; \Bbb{C})$ are K\ahler circles. In this paper we mainly study geodesic spheres in a nonflat complex space form $M^n(c; \Bbb{C})$. On these geodesic spheres we classify smooth curves whose extrinsic shapes are K\ahler circles in $M^n(c; \Bbb{C}),c\not=0$. We also give a characterization of complex space forms among K\ahler manifolds by extrinsic shapes of integral curves of characteristic vector fields on their geodesic spheres.

REAL HYPERSURFACES IN COMPLEX SPACE FORMS WITH ε-PARALLEL RICCI TENSOR AND STRUCTURE JACOBI OPERATOR

We know that there are no real hypersurfaces with parallel Ricci tensor or parallel structure Jacobi operator in a nonflat complex space form (See [4], [6], [10] and [11]). In this paper we investigate real hypersurfaces M in a nonflat complex space form Mn(c) under the condition that ∇ξS = 0 and ∇ξRξ = 0, where S and Rξ respectively denote the Ricci tensor and the structure Jacobi operator of M in Mn(c). 0. Introduction A Kaehler manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by Mn(c). It is well known that complete and simply connected complex space forms are isometric to a complex projective space PnC, a complex Euclidean space C or a complex hyperbolic space HnC according as c > 0, c = 0 and c < 0. In this paper we consider a real hypersurface M in a complex space form Mn(c), c6=0. Then M has an almost contact metric structure (φ, ξ, η, g) induced from the complex structure J and the Kaehler metric of Mn(c). The structure vector field ξ is said to be principal if Aξ = αξ is satisfied, where A denotes the shape operator of M and α = η(Aξ). A real hypersurface is said to be a Hopf hypersurface if the structure vector field ξ of M is principal. In the study of real hypersurfaces in PnC, Takagi [12] classified all homogeneous real hypersurfaces and Cecil and Ryan [2] showed that they can be regarded as the tubes of constant radius over Kaehler submanifolds when the structure vector field ξ is principal. Such tubes can be divided into six kinds of type A1, A2, B, C, D and E. Received August 8, 2005; Revised May 30, 2006. 2000 Mathematics Subject Classification. Primary 53C40; Secondary 53C15.

SOME CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE (A) IN A NONFLAT COMPLEX SPACE FORM

In this paper, we prove that if the structure Jacobi operator Rξ is ξ-parallel and Rξ commutes with the Ricci tensor S, then a real hypersurface with non-negative scalar curvature of a nonflat complex space form Mn(c) is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in Mn(c).

CHARACTERIZATIONS OF REAL HYPERSURFACES OF COMPLEX SPACE FORMS IN TERMS OF RICCI OPERATORS

We prove that a real hypersurface M in a complex space form Mn(c), <TEX>$c{\neq}0$</TEX>, whose Ricci operator and structure tensor commute each other on the holomorphic distribution and the Ricci operator is <TEX>${\eta}-parallel$</TEX>, is a Hopf hypersurface. We also give a characterization of this hypersurface.

Levi-parallel hypersurfaces in a complex space form

In this paper, we classify a Hopf hypersurface in a nonflat complex space form whose Levi-form is parallel with respect to the generalized Tanaka-Webster connection.

Real hypersurfaces of a nonflat complex space form in terms of the Ricci tensor

We know the fact that there are no real hypersurfaces with parallel Ricci tensors in a nonflat complex space form (cf. [5]). In this paper we investigate real hypersurfaces in a nonflat complex space form using some conditions of the Ricci tensor $S$ which are weaker than $\nabla S=0$. We characterize Hopf hypersurfaces of a nonflat complex space form.

THE STRUCTURE JACOBI OPERATOR ON REAL HYPERSURFACES IN A NONFLAT COMPLEX SPACE FORM

Let M be a real hypersurface with almost contact metric structure <TEX>$(\phi,\;\xi,\;\eta,\;g)$</TEX> in a nonflat complex space form <TEX>$M_n(c)$</TEX>. In this paper, we prove that if the structure Jacobi operator <TEX>$R_\xi$</TEX> commutes with both the structure tensor <TEX>$\phi$</TEX> and the Ricc tensor S, then M is a Hopf hypersurface in <TEX>$M_n(c)$</TEX> provided that the mean curvature of M is constant or <TEX>$g(S\xi,\;\xi)$</TEX> is constant.

Hopf hypersurfaces in space forms

Some geometrical properties of Hopf hypersurfaces of Kahler manifolds are introduced and a special attention is given to the case of hypersurfaces in complex projective spaces.

Real hypersurfaces in nonflat complex space forms are irreducible

The study of real hypersurfaces in complex projective space and complex hyperbolic space has been an active field over the past three decades. Although these ambient spaces might be regarded as the simplest after the spaces of constant curvature, they impose significant restrictions on the geometry of their hypersurfaces. For instance, they do not admit totally umbilical hypersurfaces and Einstein hypersurfaces. On the other hand, several important classes of real hypersurfaces in complex projective space have been constructed and investigated by many geometers. For instance, H.B. Lawson investigated real hypersurfaces of constructed by Clifford minimal hypersurfaces of +1 via Hopf fibration. R. Takagi [9] gave the list of homogeneous real hypersurfaces of . Many geometers then study the geometry from the list of Takagi and obtained various interesting geometric characterizations of homogeneous real hypersurfaces in . Another important class of real hypersurfaces in which contains the list of R. Takagi is the class of Hopf hypersurfaces. Such hypersurfaces are real hypersurfaces whose structure vector ξ is a principal curvature vector, where is the complex structure and ξ is the unit normal vector field. Examples and geometric characterizations of Hopf hypersurfaces have also been obtained by various geometers. It is known that in , is a homogeneous real hypersurface if and only if is a Hopf hypersurface with constant principal curvatures [6, 9]. The study of real hypersurfaces in complex hyperbolic space has followed developments in , often with similar results, but sometimes with differences (see [1, 7, 8] for more details). It is well-known that real projective space and real hyperbolic space admit ample hypersurfaces which are the Riemannian products of some Riemannian manifolds. It is also well-known that 3 admits a complex hypersurface which is the Riemannian

Hopf Hypersurfaces with Constant Principal Curvatures in Complex Projective or Complex Hyperbolic Spaces

A real hypersurface of a complex space form is called a Hopf hypersurface if the characteristic vector field $\xi=-JN$ on $M$ is a principal curvature vector. The main purpose of this paper is to obtain several simple geometric characterizations of all Hopf hypersurfaces with constant principal curvatures in nonflat complex space forms.

Minimality and harmonicity for Hopf vector fields

We determine when the Hopf vector fields on orientable real hypersurfaces $(M,g)$ in complex space forms are minimal or harmonic. Furthermore, we determine when these vector fields give rise to harmonic maps from $(M,g)$ to the unit tangent sphere bundle $(T_1M,g_S)$. In particular, we consider the special case of Hopf hypersurfaces and of ruled hypersurfaces. The Hopf vector fields on Hopf hypersurfaces with constant principal curvatures provide examples. The minimal ruled real hypersurfaces form another class of particular examples.

On the global structure of Hopf hypersurfaces in a complex space form

It is known that a tube over a Kähler submanifold in a complex space form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed $C^{2n-1}$ regular Hopf hypersurface in the complex projective space is a tube over an irreducible algebraic variety. In the complex hyperbolic space a connected compact generic immersed $C^{2n-1}$ regular Hopf hypersurface is a geodesic hypersphere.

Divergence-preserving geodesic transformations

We discuss divergence- and volume-preserving geodesic transformations with respect to submanifolds and in particular, with respect to hypersurfaces. We use these transformations to derive characterisations of special classes of hypersurfaces such as isoparametric hypersurfaces and Hopf hypersurfaces with constant principal curvatures. Furthermore, we consider divergence-preserving geodesic transformations with respect to geodesic spheres.

Curvature surfaces of Hopf hypersurfaces in complex space forms

We study the principal curvatures of a Hopf hypersurfaceM in ℂP n or ℂH n . The respective eigenspaces of the shape operator often turn out to induce totally real foliations ofM, whose leaves are spherical in the ambient space. Finally we classify the Hopf hypersurfaces with three distinct principal curvatures satisfying a certain non-degeneracy condition.

Almost complex curves and Hopf hypersurfaces in the nearly K�hler 6-sphere

We characterize Hopf hypersurfaces inS6 as open parts of geodesic hyperspheres or of tubes around almost complex curves ofS6.