Homogeneous Real Hypersurfaces Research Articles

A new example of a uniformly Levi degenerate hypersurface in C3

We present a homogeneous real analytic hypersurface in C3, two-nondegenerate, uniformly Levi degenerate of rank one, with a seven-dimensional CR automorphism group such that the isotropy group of each point is two-dimensional and commutative. The classical tube ΓC over the two-dimensional real cone in R3 is also homogeneous and has a seven-dimensional CR automorphism group. However, our example is not biholomorphic to the tube over the real cone, because the two-dimensional isotropy groups of ΓC are, in contrast, noncommutative.

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

Naturally reductive homogeneous real hypersurfaces in quaternionic space forms

AbstractWe determine the naturally reductive homogeneous real hypersurfaces in the family of curvature-adapted real hypersurfaces in quaternionic projective space HPn(n ≥ 3). We conclude that the naturally reductive curvature-adapted real hypersurfaces in HPn are Q-quasiumbilical and vice-versa. Further, we study the same problem in quaternionic hyperbolic space HHn(n ≥ 3).

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

We characterize homogeneous real hypersurfaces M's of type (A1), (A2) and (B) of a complex projective space in the class of real hypersurfaces by studying the holomorphic distribution T0M of M.

Invariant homogeneous structures on homogeneous real hypersurfaces in a complex projective space and an odd-dimensional sphere

Invariant homogeneous structures on homogeneous real hypersurfaces in a complex projective space and an odd-dimensional sphere

Local description of homogeneous real hypersurfaces of the two-dimensional complex space in terms of their normal equations

In the paper, a classification of real hypersurfaces of the space ℂ2 that admit transitive actions of local Lie groups of holomorphic transformations is constructed.

Certain characterizations of real hypersurfaces of type $A$ in a complex space form

complex ^-dimensional Kahler manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by Mn(c). complete and simply connected complex space form consists of a complex projective space PnC, a complex Euclidean space Cn or a complex hyperbolic space HnC, according as c > 0, c = 0 or c ,,r},g) the almost contact metric structure of M induced from the almost complex structure of Mn{c) and the shape operator of M. Eigenvalues and einvectors of are called principal curvatures and principal vectors, respectively. Many differentialgeometers have studied M from various points of view. In particular, Berndt [1] and Takagi [10] investigated the homogeneity of M. According to Takagi's classificationtheorem and Bemdt's one, the principal curvatures and their multiplicitiesof homogeneous real hypersurfaces in Mn(c) are given. Moreover, it is very interesting to characterize homogeneous real hypersurfaces of Mn{c). There are many characterizations of homogeneous ones of type since these examples have a lot of beautiful geometric properties, where type means type A or A2 in PnC and type Ao,A or A2 in HnC. Okumura [8] and Montiel-Romero [7] proved the fact in PnC and HnC, respectively that M

A characterization of all homogeneous real hypersurfaces in a complex projective space by observing the extrinsic shape of geodesics

We will give a characterization of all homogeneous real hypersurfaces in a complex projective space by observing the extrinsic shape of geodesics with initial vectors lying in the maximal holomorphic subspace of the tangent space at each point.

The rigidity for real hypersurfaces in a complex projective space

The purpose of this paper is to give a rigidity theorem for real hypersurfaces in Pn(C) satisfying a certain geometric condition. Introduction. Let Pn{C) denote an n{ > 2)-dimensional complex projective space with the metric of constant holomorphic sectional curvature 4c. We proved in [4] that two isometric immersions of a {In — l)-dimensional Riemannian manifold M into Pn(C) are congruent if their second fundamental forms coincide. In general, the type number is defined as the rank of the second fundamental form. In this paper we shall give another rigidity theorem of the same type: THEOREM A. Let M be a {In — \)-dimensίonal Riemannian manifold, and i and ϊ be two isometric immersions of M into Pn{C) {n > 3). Assume that i and ΐ have a principal direction in common at each point of M, and that the type number of (M, ί) or (M, ΐ) is not equal to 2 at each point of M. Then i and i are congruent, that is, there is a unique isometry φ of Pn{C) such that φoi = i. We shall say that an isometry φ of a real hypersurface M in Pn{C) is principal if for each point p of M there exists a principal vector v at p such that the vector φ*{v) is also principal at φ{p), where φ^ denotes the differential of φ at p. Then as an application of Theorem A we have: THEOREM B. Let M be a homogeneous real hypersurface in Pn{C) {n>3). Assume that each isometry of M is principal. Then M is an orbit under an analytic subgroup of the projective unitary group PU{n+ 1). Note that all orbits in Pn{C) under analytic subgroups of the projective unitary group PU{n+ 1) are completely classified in [4]. The authors would like to express their thanks to the referee for his useful advice. 1. Preliminaries. Let M be a {2n— l)-dimensional Riemannian manifold, and i be 1980 Mathematics Subject Classification (1985 Revision). Primary 53C40; Secondary 53C15.

The classification of naturally reductive homogeneous real hypersurfaces in complex projective space

We completely classify the naturally reductive homogeneous real hypersurfaces in a complex projective space. We conclude that only homogeneous real hypersurfaces of type (A) are naturally reductive.

Homogeneous real hypersurfaces in ℂ2

Homogeneous real hypersurfaces in ℂ2

On some classes of almost contact metric manifolds

In [1] J. Berndt and L. Vanhecke introduced two classes ((£-and 93-spaces) of Riemannian manifolds which include the class of locally symmetric spaces using the properties of Jaoobi operators along geodesies. They provided some characterizations of (£-and 23-spaces and gave the classificationsfor dimensions two and three. For further developments on the two spaces, we refer to [2], [3] and [8]. Further, T. Takahashi ([19]) introduced the notion of a (Sasakian) locally ^-symmetric space which may be considered as the analogue in the almost contact metric case of locally Hermitian symmetric spaces. Also he gave examples and equivalent properties of Sasakian locally ^-symmetric spaces. For further results about the Sasakian locally ^-symmetric spaces, we refer to [5], [6]. In the present paper, we introduce in an analogous way as in [1] four classes of almost contact metric manifolds involving Sasakian locally ^-symmetric spaces. In section 2, we recall definitionsand several elementary properties of an almost contact, a contact, a if-contact metric manifold and a Sasakian manifold. In sections 3 and 4 we give the definitionsof a %^,-space, a %%-space, a %&-space and a ^-space which are almost contact metric analogues of a (£-space or a $J$-spacein the Riemannian case. We may observe that a Sasakian manifold is a £(£-spaceand at the same time a f^-space. Also we prove that a Sasakian manifold is locally ^-symmetric if and only if it is a SDS-space and at the same time a RSJ3-space. In section 5, we show that the tangent sphere bundle of a 2-dimensional Riemannian manifold is a f$-space if and only if the base manifold is flat or of constant curvature 1. Furthermore, we give some examples of almost contact metric R(S-spaces and c^-spaces. In section 6, we consider real hypersurfaces of a complex projective space CPn with FubiniStudy metric and determine £S)8-hypersurfacesof CPn. We also show that a homogeneous real hypersurface of CPn is a fK-space, and moreover, we give

Homogeneous real hypersurfaces

Homogeneous real hypersurfaces

Characterizations of real hypersurfaces of a complex hyperbolic space in terms of Ricci tensor and holomorphic distribution

Let CPn and CHn denote the complex projective n-space with constant holomorphic sectional curvature 4, and the complex hyperbolic n-space with constant holomorphic sectional curvature ―4, respectively. Let M be a real hypersurface of CPn or CHn. M has an almost contact metric structure (<fi, ?, fj,g) induced from the complex structure / of CPn or CHn. Real hypersurfaces in CPn and CHn have been studied by many authors (cf. [1], [2], [3], [11], [12], [13], [14], [15] and [17]). For real hypersurfaces in CPn, Takagi ([16]) showed that all homogeneous real hypersurfaces in CPn are realized as the tubes of constant radius over compact Hermitian symmetric spaces of rank 1 or 2 (cf. [2] and [5]). He proved that all homogeneous real hypersurfaces in CPn could be classifiedinto six types which are said to be of type Ax, A2, B, C, D and E. Kimura ([5]) also proved that a real hypersurfaces M in CPn is homogeneous if and only if M has constant principal curvatures and £is principal. Other interesting resultsin real hypersurfaces of CPn are shown by Kimura-Maeda ([8]) and Maeda-Udagawa ([10]):

The NEF Value and Defect of Homogeneous Line Bundles

Formulas for the nef value of a homogeneous line bundle are derived and applied to the classification of homogeneous spaces with positive defect and to the classification of complete homogeneous real hypersurfaces of projective space.

The nef value and defect of homogeneous line bundles

Formulas for the nef value of a homogeneous line bundle are derived and applied to the classification of homogeneous spaces with positive defect and to the classification of complete homogeneous real hypersurfaces of projective space.

A rigidity for real hypersurfaces in a complex projective space

The purpose of this paper is to give a rigidity theorem for real hypersurfaces in Pn(C) satisfying a certain geometric condition. Introduction. Let Pn{C) denote an n{ > 2)-dimensional complex projective space with the metric of constant holomorphic sectional curvature 4c. We proved in [4] that two isometric immersions of a {In — l)-dimensional Riemannian manifold M into Pn(C) are congruent if their second fundamental forms coincide. In general, the type number is defined as the rank of the second fundamental form. In this paper we shall give another rigidity theorem of the same type: THEOREM A. Let M be a {In — \)-dimensίonal Riemannian manifold, and i and ϊ be two isometric immersions of M into Pn{C) {n > 3). Assume that i and ΐ have a principal direction in common at each point of M, and that the type number of (M, ί) or (M, ΐ) is not equal to 2 at each point of M. Then i and i are congruent, that is, there is a unique isometry φ of Pn{C) such that φoi = i. We shall say that an isometry φ of a real hypersurface M in Pn{C) is principal if for each point p of M there exists a principal vector v at p such that the vector φ*{v) is also principal at φ{p), where φ^ denotes the differential of φ at p. Then as an application of Theorem A we have: THEOREM B. Let M be a homogeneous real hypersurface in Pn{C) {n>3). Assume that each isometry of M is principal. Then M is an orbit under an analytic subgroup of the projective unitary group PU{n+ 1). Note that all orbits in Pn{C) under analytic subgroups of the projective unitary group PU{n+ 1) are completely classified in [4]. The authors would like to express their thanks to the referee for his useful advice. 1. Preliminaries. Let M be a {2n— l)-dimensional Riemannian manifold, and i be 1980 Mathematics Subject Classification (1985 Revision). Primary 53C40; Secondary 53C15.

On homogeneous real hypersurfaces in a complex projective space

On homogeneous real hypersurfaces in a complex projective space