The structure of some classes of <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S ²ⁿ⁺¹ (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S ¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.     

Einstein–Weyl structures on contact metric manifolds

In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W ^± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f η) are presented.      

Holomorphically planar conformal vector fields on contact metric manifolds

We study holomorphically planar conformal vector fields (HPCV) on contact metric manifolds under some curvature conditions. In particular, we have studied HPCV fields on (i) contact metric manifolds with pointwise constant ξ-sectional curvature (under this condition M is either K-contact or V is homothetic), (ii) Einstein contact metric manifolds (in this case M becomes K contact), (iii) contact metric manifolds with parallel Ricci tensor (under this condition M is either K-contact Einstein or is locally isometric to E ⁿ⁺¹×S ⁿ (4)).     

Contact metric manifolds with <Emphasis Type="Italic">η</Emphasis>-parallel torsion tensor

We show that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some point, is a (k, μ)-contact manifold. In particular for the standard contact metric structure of the tangent sphere bundle the torsion tensor is η-parallel if and only if M is of constant curvature, in which case its associated pseudo-Hermitian structure is CR- integrable. Next we show that if the metric of a non-Sasakian (k, μ)-contact manifold (M, g) is a gradient Ricci soliton, then (M, g) is locally flat in dimension 3, and locally isometric to E ⁿ⁺¹ × S ⁿ (4) in higher dimensions.      

Certain Contact Metrics as Ricci Almost Solitons

We show that if a compact K-contact metric is a gradient Ricci almost soliton, then it is isometric to a unit sphere S ²ⁿ⁺¹. Next, we prove that if the metric of a non-Sasakian (κ, μ)-contact metric is a gradient Ricci almost soliton, then in dimension 3 it is flat and in higher dimensions it is locally isometric to E ⁿ⁺¹ ×  S ⁿ (4). Finally, a couple of results on contact metric manifolds whose metric is a Ricci almost soliton and the potential vector field is point wise collinear with the Reeb vector field of the contact metric structure were obtained.     

Standard pseudo-Hermitian structure on manifolds and seifert fibration

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function onM. AsM supports a contact form, there exists a characteristic vector field dual to the contact structure. If induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature or of nonpositive curvature . By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When 0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.Dedicated to Professor Sasao Seiya for his sixtieth birthday     

Contact metric manifolds

In this paper we study contact metric manifoldsM ²ⁿ⁺¹(, , ,g) with characteristic vector field belonging to thek-nullity distribution. Moreover we prove that there exist i) nonK-contact, contact metric manifolds of dimension greater than 3 with Ricci operator commuting with and ii) 3-dimensional contact metric manifolds with non-zero constant -sectional curvature.     

The Rough Laplacian and Harmonicity of Hopf Vector Fields

Let M be an orientable real hypersurface of a general Kähler manifold . The characteristic vector field ξ of the induced almost contact metric structure (ξ,η, g,ϕ) is also called the Hopf vector field of M. In this paper, we compute the ‘rough’ Laplacian of ξ in terms of the shape operator A and also (as a natural generalization of the contact metric case) in terms of torsion τ = L_ξ g. Then we give some criteria of harmonicity of ξ. Moreover, we consider hypersurfaces M of contact type and give some criteria for M to admit an H-contact structure.Mathematics Subject Classifications (2000): 53C25, 53C20, 53C40, 53D35.     

Almost Contact Lagrangian Submanifolds of Nearly Kaehler 6-Sphere

For a Lagrangian submanifold M of S ⁶ with nearly Kaehler structure, we provide conditions for a canonically induced almost contact metric structure on M by a unit vector field, to be Sasakian. Assuming M contact metric, we show that it is Sasakian if and only if the second fundamental form annihilates the Reeb vector field ξ, furthermore, if the Sasakian submanifold M is parallel along ξ, then it is the totally geodesic 3-sphere. We conclude with a condition that reduces the normal canonical almost contact metric structure on M to Sasakian or cosymplectic structure.     

Five-dimensional K-contact Lie algebras

We introduce a general approach to the study of left-invariant K-contact structures on Lie groups and we obtain a full classification in dimension five. We show that Sasakian structures on five-dimensional Lie algebras with non-trivial center are a relatively rare phenomenon with respect to K-contact structures. We also prove that a five-dimensional solvmanifold with a left-invariant K-contact (not Sasakian) structure is a \mathbb S¹{\mathbb S^1} -bundle over a symplectic solvmanifold. Rigidity results are then obtained for five-dimensional K-contact Lie algebras with trivial center and for K-contact η-Einstein structures. Moreover, five-dimensional Sasakian φ-symmetric Lie algebras are completely classified, and some explicit examples of five-dimensional Sasakian pseudo-metric Lie algebras are provided.     

The contact Yamabe flow on <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We use the contact Yamabe flow to find solutions of the contact Yamabe problem on K-contact manifolds.      

Torsion and conformally Anosov flows in contact Riemannian geometry

We prove that on a compact (non Sasakian) contact metric 3-manifold with critical metric for the Chern-Hamilton functional, the characteristic vector field ξ is conformally Anosov and there exists a smooth curve in the contact distribution of conformally Anosov flows. As a consequence, we show that negativity of the ξ-sectional curvature is not a necessary condition for conformal Anosovicity of ξ (this completes a result of [4]). Moreover, we study contact metric 3-manifolds with constant ξ-sectional curvature and, in particular, correct a result of [13].     

B.Y. Chen inequalities for slant submanifolds in Sasakian space forms

In the present paper, we establish Chen inequalities for slant submanifolds in Sasakian space forms, by using subspaces orthogonal to the Reeb vector field ξ.     

Almost contact metric manifolds whose Reeb vector field is a harmonic section

We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ??, when ??? is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ?? is geodesic, ?? is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures ?? are harmonic sections, in the sense of Vergara-Diaz and Wood?[25], and in some cases they are also harmonic maps.     

On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S¹ → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X^↑ project on a nonlinear system of subelliptic PDEs on M. Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20     

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.     

Biharmonic curves in 3-dimensional Sasakian space forms

We show that every proper biharmonic curve in a 3-dimensional Sasakian space form of constant holomorphic sectional curvature H is a helix (both of whose geodesic curvature and geodesic torsion are constants). In particular, if H ≠  1, then it is a slant helix, that is, a helix which makes constant angle α with the Reeb vector field with the property . Moreover, we construct parametric equations of proper biharmonic herices in Bianchi–Cartan–Vranceanu model spaces of a Sasakian space form.      

Quaternionic and para-quaternionic <Emphasis Type="Italic">CR</Emphasis> structure on (4n+3)-dimensional manifolds

We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω₁,ω₂,ω₃) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described.     

On Ricci curvature ofC-totally real submanifolds in Sasakian space forms

LetM ⁿ be a Riemanniann-manifold. Denote byS(p) and Ric(p) the Ricci tensor and the maximum Ricci curvature onM _n, respectively. In this paper we prove that everyC-totally real submanifold of a Sasakian space formM ^2m+1(c) satisfies , whereH ² andg are the square mean curvature function and metric tensor onM ⁿ, respectively. The equality holds identically if and only if eitherM ⁿ is totally geodesic submanifold or n = 2 andM ⁿ is totally umbilical submanifold. Also we show that if aC-totally real submanifoldM ⁿ ofM ²ⁿ⁺¹ (c) satisfies identically, then it is minimal.     

Normal CR structures on compact 3-manifolds

We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotients of or of a non-flat circle bundle over a Riemann surface of positive genus. In the latter case, we prove that their CR automorphisms group is a finite extension of , and we classify the normal CR structures on these manifolds. Received: 14 March 2000 / Published online: 17 May 2001