Symplectic manifolds with contact type boundaries

Summary An example of a 4-dimensional symplectic manifold with disconnected boundary of contact type is constructed. A collection of other results about symplectic manifolds with contact-type boundaries are derived using the theory ofJ-holomorphic spheres. In particular, the following theorem of Eliashberg-Floer-McDuff is proved: if a neighbourhood of the boundary of (V, ) is symplectomorphic to a neighbourhood ofS ^2n–1 in standard Euclidean space, and if vanishes on all 2-spheres inV, thenV is diffeomorphic to the ballB ²ⁿ.Oblatum 19-III-1990Partially supported by NSF grant no: DMS 8803056     

On contact metric statistical manifolds

A notion of almost contact metric statistical structure is introduced and thereby contact metric and K-contact statistical structures are defined. Furthermore a necessary and sufficient condition for a contact metric statistical manifold to admit K-contact statistical structure is given. Finally, the condition for an odd-dimensional statistical manifold to have K-contact statistical structure is expressed.     

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.     

On contact strongly pseudo-convex integrableCR manifolds

It is shown that a locally symmetric contact strongly pseudo-convex integrableCR manifold of dimension greater than 3 and other than 7 is locally isometric to a unit sphere or the Riemannian product of an (n + 1)-dimensional Euclidean space and a sphere. A conformally flat contact strongly pseudo-convex integrableCR manifold is locally isometric to a unit sphere, provided the characteristic vector field is an eigenvector of the Ricci tensor at each point.     

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 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 the curvature of a generalization of contact metric manifolds

Summary We consider a genaralization of contact metric manifolds given by assignment of 1-formsη¹, . . . ,η^sand a compatible metric gon a manifold. With some integrability conditions they are called almost<span style='font-size:10.0pt;font-family:"Monotype Corsiva"; mso-bidi-font-family:"Monotype Corsiva"'>S-manifolds. We give a sufficient condition regarding the curvature of an almost<span style='font-size:10.0pt;font-family:"Monotype Corsiva";mso-bidi-font-family: "Monotype Corsiva"'>S-manifold to be locally isometric to a product of a Euclidean space and a sphere.     

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.     

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 classification of almost contact metric manifolds

On connected manifolds of dimension higher than three, the non-existence of 132 Chinea and González-Dávila types of almost contact metric structures is proved. This is a consequence of some interrelations among components of the intrinsic torsion of an almost contact metric structure. Such interrelations allow to describe the exterior derivatives of some relevant forms in the context of almost contact metric geometry.     

On complete manifolds supporting a weighted Sobolev type inequality

This paper studies the geometric and topological properties of complete open Riemannian manifolds which support a weighted Sobolev or log-Sobolev inequality. We show that the constant in the weighted Sobolev inequality on a complete open Riemannian manifold should be bigger than or equal to the optimal one on the Euclidean space of the same dimension and that a complete open manifold of asymptotically non-negative Ricci curvature supporting a weighted Sobolev inequality must have large volume growth. We also show that a complete manifold of non-negative Ricci curvature on which the log-Sobolev inequality holds is not very far from the Euclidean space.     

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     

Bifurcations on contact manifolds

Consider a 1-parameter compactly supported family of Legendrian submanifolds of the 1-jet bundle of a compact manifold with its natural contact structure and a path of intersection points of the Legendrian family with the 1-jet of a constant function. Since the contact distribution is a symplectic vector bundle, it is possible to assign a Maslov-type index to the intersection path. We show that the non-vanishing of the Maslov intersection index implies that there exists at least one point of bifurcation from the given path of intersection points. This result can be viewed as a kind of analogue in bifurcation theory of the Arnold-Sandon conjecture on intersections of Legendrian submanifols. The proof is based on the technique of generating functions that relates the properties of Hamiltonian diffeomorphisms to the Morse theory of the associated functions.     

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.     

On twisted contact groupoids and on integration of twisted Jacobi manifolds

We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds with twisted Jacobi manifolds and homogeneous twisted symplectic groupoids with twisted contact ones. Some examples for each structure are presented.     

On almost pseudo symmetric manifolds admitting a type of semi-symmetric non-metric connection

The object of the present paper is to study almost pseudo symmetric manifolds admitting a type of semi-symmetric non-metric connection. Also we consider a special conformally flat almost pseudo symmetric manifold admitting a type of semi-symmetric non-metric connection.