Generalized Cauchy-Riemann lightlike submanifolds of indefinite Sasakian manifolds

We study a class of submanifolds, called Generalized Cauchy-Riemann (GCR) lightlike submanifolds of indefinite Sasakian manifolds as an umbrella of invariant, screen real, contact CR lightlike subcases [8] and real hypersurfaces [9]. We prove existence and non-existence theorems and a characterization theorem on minimal GCR-lightlike submanifolds.     

On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds

We study semi-parallel lightlike hypersurfaces of an indefinite Kenmotsu manifold, tangent to the structure vector field. Some Theorems on parallel and semi-parallel vector field, geodesibility of lightlike hypersurfaces are obtained. The geometrical configuration of such lightlike hypersurfaces is established. We prove that, in totally contact umbilical lightlike hypersurfaces of an indefinite Kenmotsu manifold which has constant ${\overline{\phi}}$ -holomorphic sectional curvature c, tangent to the structure vector field and such that its distribution is parallel, the parallelism and semi-parallelism notions are equivalent.     

Lightlike real hypersurfaces of indefinite quaternion Kaehler manifold

In this paper, we introduce real lightlike hypersurfaces of indefinite quaternion Kaehler manifold. Fundamental properties of real lightlike hypersurfaces of an indefinite quaternion Kaehler manifold are investigated. We prove the non existence of real lightlike hypersurfaces in indefinite qaternionic space form under some conditions. Received 31 October 2000; revised 20 June 2001.     

Einstein-Weyl structures on lightlike hypersurfaces

We study Weyl structures on lightlike hypersurfaces endowed with a conformal structure of certain type and specific screen distribution: the Weyl screen structures. We investigate various differential geometric properties of Einstein-Weyl screen structures on lightlike hypersurfaces and show that, for ambient Lorentzian space ? ₁ ⁿ⁺² and a totally umbilical screen foliation, there is a strong interplay with the induced (Riemannian) Weyl-structure on the leaves.     

On lightlike geometry in indefinite Kenmotsu manifolds

We investigate some geometric aspects of lightlike hypersurfaces of indefinite Kenmotsu manifolds, tangent to the structure vector field, by paying attention to the geometry of leaves of integrable distributions. Theorems on parallel vector fields, Killing distribution, geodesibility of their leaves are obtained. The geometric configuration of such lightlike hypersurfaces and leaves of its screen integrable distributions are established. We show that no totally contact umbilical leaf of a screen integrable distribution of a lightlike hypersurface can be an extrinsic sphere. We also prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle.     

Biharmonic hypersurfaces in Sasakian space forms

We consider the Boothby–Wang fibration of a strictly regular Sasakian space form N and find the characterization of biharmonic Hopf cylinders over submanifolds of . Then, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces.     

On Positive Sasakian Geometry

A Sasakian structure =(\xi,\eta,\Phi,g) on a manifold Mis called positiveif its basic first Chern class c₁( ) can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This provides us with a new technique for proving the existence of positive Ricci curvature metrics on certain odd dimensional manifolds. As an example we give a completely independent proof of a result of Sha and Yang that for every nonnegative integer kthe 5-manifolds k#(S ²×S ³) admits metrics of positive Ricci curvature.     

Foliations of lightlike hypersurfaces and their physical interpretation

This paper deals with a family of lightlike (null) hypersurfaces (H _u ) of a Lorentzian manifold M such that each null normal vector ℓ of H _u is not entirely in H _u , but, is defined in some open subset of M around H _u . Although the family (H _u ) is not unique, we show, subject to some reasonable condition(s), that the involved induced objects are independent of the choice of (H _u ) once evaluated at u = constant. We use (n+1)-splitting Lorentzian manifold to obtain a normalization of ℓ and a well-defined projector onto H, needed for Gauss, Weingarten, Gauss-Codazzi equations and calculate induced metrics on proper totally umbilical and totally geodesic H _u . Finally, we establish a link between the geometry and physics of lightlike hypersurfaces and a variety of black hole horizons.     

Radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric

In this paper we introduce radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric. Such class of lightlike hypersurfaces cannot exist for indefinite almost Hermitian manifolds. The considered lightlike hypersurfaces have two important properties. The first one is the uniqueness of their screen distributions, which implies that the induced geometric objects are well-defined. The second property is that the induced Ricci tensor on radical transversal lightlike hypersurface of a Kähler manifold with Norden metric is symmetric. This allows to define an induced scalar curvature of the hypersurface. We obtain new results about lightlike hypersurfaces concerning their relations with non-degenerate hypersurfaces of almost complex manifolds with Norden metric. Examples of the considered hypersurfaces are given.     

Spherical-type hypersurfaces in a Riemannian manifold

We extend some rigidity results of Aleksandrov and Ros on compact hypersurfaces inR ⁿ to more general ambient spaces with the aid of the notion of almost conformal vector fields. These latter, at least locally, always exist and allow us to find interesting integral formulas fitting our purposes.     

Geometry of tubes about characteristic curves on Sasakian manifolds

One studies, using Riemannian foliation theory, some aspects of the intrinsic and extrinsic geometry of small tubes about the flow lines of the characteristic vector field on a Sasakian manifold. In particular, one focuses on some characteristic properties of the shape operator and the Ricci operator of these tubes for the classes of ?-symmetric spaces and Sasakian space forms.     

Congruence theorems for compact hypersurfaces of a riemannian manifold

Summary Let M^m, ^m be two m-dimensional compact oriented hypersurfaces of class C³ immersed in a Riemannian manifold R^m+1 of constant sectional curvature. Suppose that R^m+1 admits a one-parameter continuous group G of conformal transformations satisfying a certain condition (which holds automatically when G is a group of isometric transformations). Suppose further that there is a1 − 1 transformation T_τ ∈ G between M^m and ^m such that for each P ∈ M^m and each ^m. If the r-th mean curvature for any r, 1 ⩽ r ⩽ m, of M^m at each point P ∈ M^m is equal to that of ^m at the corresponding point , together with other conditions, then M^m and ^m are congruent mod G. This is a generalization of a joint theorem ofH. Hopf andY. Katsurada [5] in which G is a group of isometric transformations. Entrata in Redazione il 13 Giugno 1975. The first author was partially supported by the National Science Foundation grant GP-33944.     

Geometry on the quantum Heisenberg manifold

A class of C∗-algebras called quantum Heisenberg manifolds were introduced by Rieffel in (Comm. Math. Phys. 122 (1989) 531) as strict deformation quantization of Heisenberg manifolds. Using the ergodic action of Heisenberg group we construct a family of spectral triples. It is shown that associated Kasparov modules are homotopic. We also show that they induce cohomologous elements in entire cyclic cohomology. The space of Connes-deRham forms have been explicitly calculated. Then we characterize torsionless/unitary connections and show that there does not exist a connection that is simultaneously torsionless and unitary. Explicit examples of connections are produced with negative scalar curvature. This part illustrates computations involving some of the concepts introduced in Frohlich et al. (Comm. Math. Phys. 203 (1999) 119), for which to the best of our knowledge no infinite-dimensional example is known other that the noncommutative torus.     

Complete harmonic stable minimal hypersurfaces in a Riemannian manifold

In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative Ricci curvature is conformally equivalent to either a plane R ² or a cylinder R × S ¹, which generalizes a theorem due to Fischer-Colbrie and Schoen [12]. The second one is that an n ≥ 2-dimensional, complete harmonic stable minimal, hypersurface M in a complete Riemannian manifold with non-negative sectional curvature has only one end if M is non-parabolic. The third one, which we prove, is that there exist no non-trivial L ²-harmonic one forms on a complete harmonic stable minimal hypersurface in a complete Riemannian manifold with non-negative sectional curvature. Since the harmonic stability is weaker than stability, we obtain a generalization of a theorem due to Miyaoka [20] and Palmer [21]. Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The author’s research was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation, Korea. Authors’ addresses: Qing-Ming Cheng, Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan; Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea     

Geometry of the cotangent bundle with Sasakian metrics and its applications

The main aim of this paper is to study paraholomorpic Sasakian metric and Killing vector field with respect to the Sasakian metric in the cotangent bundle.