On the deformation theory of Calabi-Yau structures in strongly pseudo-convex manifolds

We study the deformation theory of Calabi-Yau structures in strongly pseudo-convex manifolds with trivial canonical bundles. Our approach could be considered as an alternative proof for a theorem of H. Laufer on the deformation of strongly pseudo-convex surfaces.     

On certain non-Kahlerian strongly pseudoconvex manifolds

It has been conjectured that strongly pseudoconvex manifoldsX such that its exceptional setS is an irreducible curve can be embedded biholomorphically into some ? ^N ×P _m . In this paper we show that this is true, with one exception, namely when dim_? X = 3 and its first Chern classc ₁ (K _X ¦S) = 0 whereS ?P ₁ andK _X is the canonical bundle ofX. On the other hand, we explicitly exhibit such a 3-foldX that is not Kahlerian; also we construct non-Kahlerian strongly pseudoconvex 3-foldX whose exceptional setS is a ruled surface; those concrete examples naturally raise the possibility of classifying non-Kahlerian strongly pseudoconvex 3-folds.     

On conformally flat contact metric manifolds

In the present paper we classify the conformally flat contact metric manifolds of dimension satisfying . We prove that these manifolds are Sasakian of constant curvature 1.     

On 2-almost contact tachibana manifolds

LeM be a (2m+2)-dimensional Riemannian manifold with two structure vector fieldsξ _r (r=2m+1, 2m+2) and letη ^r =ξ _r ^b be their corresponding covectors (or Pfaffians). These vector fields define onM a 2-almost contact structure. If the 2-formϕ=η ^2m+1∧η ^2m+2 is harmonic, then, following S. Tachibana [12],M is a Tachibana manifold and in this caseM is covered with 2 families of minimal submanifolds tangent toD ^⊥={ξ r} and its complementary orthogonal distributionD ^⊤. On such a manifold a canonical eigenfunction α of the Laplacian is associated. Since the corresponding eingenvalue is negative,M cannot be compact. Any horizontal vector fieldX orthogonal to α^# is a skew-symmetric Killing vector field (see [6]). Next, we assume that the Tachibana manifoldM under consideration is endowed with a framedf-structure defined by an endomorphism ϕ of the tangent bundleTM. Infinitesimal automorphisms of the symplectic form Ω _ϕ are obtained.     

On the quasi-projectivity of compactifiable strongly pseudoconvex manifolds

We construct new examples of non-Kahlerian 1-convex threefolds X with exceptional set≅P₁ (resp. ≅F₂). Also the structure of Pic(X) will be studied. On the other hand, we shall investigate the quasi-projective structure of certain Kahlerian compactifiable 1-convex manifolds; particular attention will be given to 3-fold cases through concrete examples.     

On strongly pseudoconvex manifolds with one-dimensional exceptional set

Let X be a strongly pseudoconvex n-dimensional manifold with one-dimensional connected exceptional set S. Here we show that if S is reducible, then X is embeddable into C^N ×P ^{m for some N, m and in particular X is Kählerian with a possible exception for n = 3; we analyze this exceptional case but we do not know if it may occur. The case in which S is irreducible was previously analyzed by Tan [11], who proved that X is embeddable if either n ≠ 3 or S is not isomorphic to} P ¹. In the same paper, Tan gave an example of a non-embeddable and non-Kählerian strongly pseudoconvex three-dimensional manifold withP ¹ as an exceptional set.     

On some classes of contact metric manifolds

We define four new classes of contact metric manifoulds using Tanaka connection and Jacobi operators. We prove that a contact metric manifold with the structure vector field ξ belonging to thek-nullity distribution is contact metric locally ?-symmetric (in the sense of D. B. Blair) if and only if the manifold is a and space. Also, we prove that a 3-dimensional contact metric and is locally ?-symmetric (in the sense of D. E. Blair) and give counter-examples of the converse.     

On the curvature of contact metric manifolds

Some results on Ricci-symmetric contact metric manifolds are obtained. Second order parallel tensors and vector fields keeping curvature tensor invariant are characterized on a class of contact manifolds. Conformally flat contact manifolds are studied assuming certain curvature conditions. Finally some results onk-nullity distribution of contact manifolds are obtained.     

Projective contact manifolds

The present work is concerned with the study of complex projective manifolds X which carry a complex contact structure. In the first part of the paper we show that if K _X is not nef, then either X is Fano and b ₂(X)=1, or X is of the form ℙ(T _Y ), where Y is a projective manifold. In the second part of the paper we consider contact manifolds where K _X is nef. Oblatum 15-X-1999 & 3-II-2000?Published online: 8 May 2000     

On a class of 3-dimensional contact metric manifolds

Let M³ be a 3-dimensional contact metric manifold with contact structure (, , , g), such that and =R(.,)) commute. Such a manifold is called 3--manifold. We prove that every 3--manifold with -parallel Weyl tensor is either flat or a Sasakian manifold with constant curvature 1.     

Levi-parallel contact Riemannian manifolds

We study the Riemannian geometry of contact manifolds with respect to a fixed admissible metric, making the Reeb vector field unitary and orthogonal to the contact distribution, under the assumption that the Levi–Tanaka form is parallel with respect to a canonical connection with torsion.