Taut contact hyperbolas on three-manifolds

Annals of Global Analysis and Geometry - In this paper, we introduce the notion of taut contact hyperbola on three-manifolds. It is the hyperbolic analogue of the taut contact circle notion...     

Special directions on contact metric three-manifolds

Blair [5] has introduced special directions on a contact metric 3-manifolds with negative sectional curvature for plane sections containing the characteristic vector field and, when is Anosov, compared such directions with the Anosov directions. In this paper we introduce the notion of Anosov-like special directions on a contact metric 3-manifold. Such directions exist, on contact metric manifolds with negative -Ricci curvature, if and only if the torsion is -parallel, namely (1.1) is satisfied. If a contact metric 3-manifold M admits Anosov-like special directions, and is -parallel, where is the Berger-Ebin operator, then is Anosov and the universal covering of M is the Lie group (2,R). We note that the notion of Anosov-like special directions is related to that of conformally Anosow flow introduced in [9] and [14] (see [6]).Supported by funds of the M.U.R.S.T. and of the University of Lecce. 1991.     

Torsion and homogeneity on contact metric three-manifolds

We show that a three-dimensional contact metric manifold is locally homogeneous if and only if it is ball-homogeneous and satisfies the condition ∇_ξτ=2aτϕ, with a constant. Then, we relate the condition ∇_ξτ=0 with the existence of taut contact circles on a compact three-dimensional contact metric manifold. Entrata in Redazione il 20 gennaio 1999. Supported by funds of the University of Lecce and the M.U.R.S.T. Work made within the program of G.N.S.A.G.A.-C.N.R.     

Fillable contact structures on torus bundles over circles

We will construct an example of a strongly symplectically fillable contact structure on a torus bundle over the circle with parabolic monodromy.

     

A variational characterization of contact metric structures

Annals of Global Analysis and Geometry - On a manifold with a given nowhere vanishing vector field, we examine the squared $$L^2$$ -norm of the integrability tensor of the orthogonal complement of...     

Quadratic functions and complex spin structures on three-manifolds

We show how the space of complex spin structures of a closed oriented three-manifold embeds naturally into a space of quadratic functions associated to its linking pairing. Besides, we extend the Goussarov-Habiro theory of finite type invariants to the realm of compact oriented three-manifolds equipped with a complex spin structure. Our main result states that two closed oriented three-manifolds endowed with a complex spin structure are undistinguishable by complex spin invariants of degree zero if, and only if, their associated quadratic functions are isomorphic.     

Invariants of conformal transformations of almost contact metric structures

The so-called structure tensors of almost contact metric structures, which play a key role in the geometry of almost contact metric structures, are explicitly calculated. The transformations of these tensors under conformal transformations of almost contact metric structures are described. The results obtained are used to study the behavior of the most interesting classes of almost contact structures under conformal transformations.     

Examples of a generalization of contact metric structures on fibre bundles

We consider a Riemannian manifold with a compatible f-structure which admits a parallelizable kernel. With some additional integrability conditions it is called (almost) -manifold and is a natural generalization of the (almost) contact metric and the Sasakian manifolds. There are presented various methods of constructing examples of such manifolds. There are used structures on the principal bundles and the pull-back bundles. Then there are considered relations between (almost) -manifolds and transverse almost Hermitian structures on the foliated manifolds. Research supported by the Italian MIUR 60% and GNSAGA.     

Almost contact Riemannian three-manifolds with Reeb flow symmetry

In this paper we give a complete classification of simply connected homogeneous almost α-Kenmotsu three-manifolds M whose Ricci operator is invariant along the Reeb flow. We get this classification by using the Gaussian and the extrinsic curvature associated with the canonical foliation of M.     

Homogeneity on three-dimensional contact metric manifolds

We study ball-homogeneity, curvature homogeneity, natural reductivity, conformal flatness and ϕ-symmetry for three-dimensional contact metric manifolds. Several classification results are given. Member of G.N.S.A.G.A. Supported by funds of the M.U.R.S.T.     

A dynamical characterization of contact circles

A simple dynamical condition is given for line fields within a contact structure, which is satisfied exactly by those line fields which are common kernels of contact circles. Some convexity properties are established which are useful in the study of contact circles up to homotopy. A necessary condition for extending a contact form to a contact circle is given. Based on work of Lisca–Matić and Honda, concrete examples are given which show that the condition is not just homotopic but a geometric one. The paper ends with an open question. Dedicated to Fernando Varela, for many reasons.     

Almost contact metric submersions

In this paper we start with a brief survey on the theory of Riemannian submersions between almost contact metric manifolds.     

Harmonicity on some almost contact metric manifolds

In this paper we shall study some classes of harmonic maps on Trans-Sasakian manifolds.     

Vassiliev invariants of Legendrian, transverse, and framed knots in contact three-manifolds

We show that for a large class of contact three-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's J⁺-type invariants of wave fronts on a surface F is isomorphic to the group of Vassiliev invariants of framed knots in the spherical cotangent bundle ST∗F of F.On the other hand, we construct the first examples of contact manifolds for which Vassiliev invariants of Legendrian knots can distinguish Legendrian knots that realize isotopic framed knots and are homotopic as Legendrian immersions.     

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.