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.     

Left invariant contact structures on Lie groups

Amongst other results, we perform a ‘contactization’ method to construct, in every odd dimension, many contact Lie groups with a discrete center, unlike the usual (classical) contactization which only produces Lie groups with a non-discrete center. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. We give classification results in low dimensions. A complete list is supplied in dimension 5. In any odd dimension greater than 5, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also characterize solvable contact Lie algebras whose derived ideal has codimension one. For simplicity, most of the results are given in the Lie algebra version.     

The geometry of a bi-Lagrangian manifold

This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations. We also study the non-integrable case (i.e., a symplectic manifold endowed with two transversal Lagrangian distributions). We show that many different geometric structures can be attached to these manifolds and we carefully analyze the associated connections. Moreover, we introduce the problem of the intersection of the two leaves, one of each foliation, through a point and show a lot of significative examples.     

η-parallel contact metric spaces

We prove that a contact metric manifold M=(M;η,ξ,φ,g) with η-parallel tensor h is either a K-contact space or a (k,μ)-space, where h denotes, up to a scaling factor, the Lie derivative of the structure tensor φ in the direction of the characteristic vector ξ. In the latter case, its associated CR-structure is in particular integrable.     

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.     

A nullity condition for complex contact metric manifolds

A nullity condition for real contact manifolds is defined by Blair, Koufogiorgos and Papantoniu. Lately, Boeckx classified such manifolds completely. In this paper, a nullity condition for complex contact manifolds is defined as follows: take a complex contact manifold whose vertical space is annihilated by the curvature. Then, apply an $\mathcal{H}$-homothetic deformation. In this way, we get a condition which is invariant under $\mathcal{H}$-homothetic deformations. A complex contact manifold satisfying this condition is called a complex (,)-space. Some curvature properties of complex (,)-spaces are studied and it is shown that, just as in the real case, the curvature tensor of a complex (,)-space is completely determined.     

Two classes of conformally flat contact metric 3-manifolds

We give a classification of 3—dimensional conformally flat contact metric manifolds satisfying: =0(=L g) orR(Y, Z)=k[(Z)Y–(Y)Z]+[(Z)hY]–(Y)hZ] wherek and are functions. It is proved that they are flat (the non-Sasakian case) or of constant curvature 1 (the Sasakian case).     

Eigenvalue estimate of the basic Dirac operator on a Kähler foliation

Let F be a Kähler spin foliation of codimension q=2n on a compact Riemannian manifold M with the transversally holomorphic mean curvature form κ. It is well known [S.D. Jung, T.H. Kang, Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation, J. Geom. Phys. 45 (2003) 75-90] that the eigenvalue λ of the basic Dirac operator Db satisfies the inequality , where σ∇ is the transversal scalar curvature of F. In this paper, we introduce the transversal Kählerian twistor operator and prove that the same inequality for the eigenvalue of the basic Dirac operator by using the transversal Kählerian twistor operator. We also study the limiting case. In fact, F is minimal and transversally Einsteinian of odd complex codimension n with nonnegative constant transversal scalar curvature.     

Branched spines and contact structures on 3-manifolds

We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover we show that global uniqueness holds when restricting to tight structures. We establish branched versions of the elimination lemma. We prove a smooth version of the Gillman-Rolfsen PL-embedding theorem, deducing that branched spines can be used to construct contact structures in a given homotopy class of plane fields. Entrata in Redazione il 6 novembre 1998.     

Good coverings for section spaces of fibre bundles

In this note, existence of good coverings for section spaces of smooth fibre bundles is shown by resorting to the Riemannian geometry of the L²-metric, via the construction of strongly convex neighbourhoods, in close analogy with the finite dimensional case.     

Examples of almost Einstein structures on products and in cohomogeneity one

Almost Einstein manifolds are conformally Einstein up to a scale singularity, in general. This notion comes from conformal tractor calculus. In the current paper we discuss almost Einstein structures on closed Riemannian product manifolds and on 4-manifolds of cohomogeneity one. Explicit solutions are found by solving ordinary differential equations. In particular, we construct three families of closed 4-manifolds with almost Einstein structure corresponding to the boundary data of certain unimodular Lie groups. Two of these families are Bach-flat, but neither (globally) conformally Einstein nor half conformally flat. On products with a 2-sphere we find an exotic family of almost Einstein structures with hypersurface singularity as well.     

Tight contact structures with no symplectic fillings

We exhibit tight contact structures on 3-manifolds that do not admit any symplectic fillings. Oblatum 7-XII-2000 & 14-XI-2001?Published online: 9 April 2002     

A splitting theorem for compact complex homogeneous spaces with a symplectic structure

In this note we prove a splitting theorem for compact complex homogeneous spaces with a cohomology 2 class [] such that the top power [ ⁿ ]0.Dedicated to Professor W. C. Hsiang on the occasion of his 60th birthdayPartially supported by NSF Grant DMS-9401755.     

Almost contact metric 5-manifolds and connections with torsion

We study 5-dimensional Riemannian manifolds that admit an almost contact metric structure. We classify these structures by their intrinsic torsion and review the literature in terms of this scheme. Moreover, we determine necessary and sufficient conditions for the existence of metric connections with vectorial, totally skew-symmetric or traceless cyclic torsion that are compatible with the almost contact metric structure. Finally, we examine explicit examples of almost contact metric 5-manifolds from this perspective.     

On vertical skew symmetric almost contact 3-structures

We consider a (2m + 3)-dimensional Riemannian manifold M(ξ _r, η^r, g ) endowed with a vertical skew symmetric almost contact 3-structure. Such manifold is foliated by 3-dimensional submanifolds of constant curvature tangent to the vertical distribution and the square of the length of the vertical structure vector field is an isoparametric function. If, in addition, M(ξ _r, η^r, g ) is endowed with an f -structure φ, M, turns out to be a framed f−CR-manifold. The fundamental 2-form Ω associated with φ is a presymplectic form. Locally, M is the Riemannian product of two totally geodesic submanifolds, where is a 2m-dimensional Kaehlerian submanifold and is a 3-dimensional submanifold of constant curvature. If M is not compact, a class of local Hamiltonians of Ω is obtained.     

Contact structures, equivariant spin bordism, and periodic fundamental groups

Using contact surgery and equivariant bordism theory, we prove the existence of contact structures on all 5-dimensional spin manifolds with certain finite fundamental groups. Received September 13, 1999 / Revised version September 13, 2000 / Published online April 12, 2001