Elliptic open books on torus bundles over the circle

As an application of the construction of open books on plumbed 3-manifolds, we construct elliptic open books on torus bundles over the circle. In certain cases these open books are compatible with Stein fillable contact structures and have minimal genus.      

Poisson fiber bundles and coupling Dirac structures

Poisson fiber bundles are studied. We give sufficient conditions for the existence of a Dirac structure on the total space of a Poisson fiber bundle endowed with a compatible connection. We also provide some examples.      

Optimal packings of two to four equal circles on any flat torus

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.     

On 4-manifolds homotopy equivalent to surface bundles over surfaces

We show that a closed 4-manifold is homotopy equivalent to the total space of a surface bundle over a surface if the obviously necessary conditions on the fundamental group and Euler characteristic hold. When the base is the 2-sphere we need also conditions on the characteristic classes of the manifold. (Our results are incomplete when the base is the projective plane.) In most cases we can show the manifold is s-cobordant to the total space of the bundle.     

Taut foliations in punctured surface bundles, I

Given a fibred, compact, orientable 3-manifold with single boundarycomponent, we show that a fibration with fiber surface of negativeEuler characteristic can be perturbed to yield taut foliationsrealizing an open interval of boundary slopes about the boundaryslope of the fibration. These taut foliations extend to tautfoliations in the corresponding surgery manifolds. 2000 MathematicsSubject Classification: primary 57M25; secondary 57R30.     

Some liftings of poisson structures to Weil bundles

We establish a formula for the Schouten-Nijenhuis bracket of linear liftings of skew-symmetric tensor fields to any Weil bundle. As a result we obtain a construction of some liftings of Poisson structures to Weil bundles.     

Relative connections on principal bundles and relative equivariant structures

We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over a complex analytic family. We also introduce the notion of relative equivariant bundles and establish its relation with relative holomorphic connections on principal bundles.     

Real Places and Torus Bundles

If M is a hyperbolic once-punctured torus bundle over S ¹, then the trace field of M has no real places.     

Classification of tight contact structures on small Seifert 3-manifolds with

We classify positive, tight contact structures on closed Seifert fibered 3-manifolds with base , three singular fibers and .

     

Flat holomorphic connections on principal bundles over a projective manifold

Let be a connected complex linear algebraic group and its unipotent radical. A principal -bundle over a projective manifold will be called polystable if the associated principal -bundle is so. A -bundle over is polystable with vanishing characteristic classes of degrees one and two if and only if admits a flat holomorphic connection with the property that the image in of the monodromy of the connection is contained in a maximal compact subgroup of .

     

On almost complex structures on tangent bundles

Summary Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(M,J,g)$ be a K\&quot;ahler--Norden manifold. Using the notions of the horizontal and vertical lifts, a class of almost complex structures $\widetilde J$ is defined on the tangent bundle $T\!M$, and necessary and sufficient conditions for such a structure to be integrable (complex) are described. Next, a class of pseudo-Riemannian metrics $\widetilde g$ of Norden type is defined on $T\!M$, for which $\widetilde J$ is an antiisometry. Thus, the pair $(\widetilde J,\widetilde g)$ becomes an almost complex structure with Norden metric on $T\!M$. It is checked whether the structure $(\widetilde J,\widetilde g)$ is K\&quot;ahler--Norden itself.     

A note on the Picard bundle over a moduli space of vector bundles

Let ??(n , d ) be a coprime moduli space of stable vector bundles of rank n ≥ 2 and degree d over a complex irreducible smooth projective curve X of genus g ≥ 2 and ??_ξ ? ??(n , d ) a fixed determinant moduli space. Assuming that the degree d is sufficiently large, denote by ?? the vector bundle over X ×??(n , d ) defined by the kernel of the evaluation map H ⁰(X , E ) → E_x , where E ∈??(n , d ) and x ∈ X . We prove that ?? and its restriction ??_ξ to X × ??ξ are stable. The space of all infinitesimal deformations of ?? over X ×??(n , d ) is proved to be of dimension 3g and that of ??ξ over X × ??ξ of dimension 2g , assuming that g ≥ 3 and if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the classification of contact metric ‐spaces via tangent hyperquadric bundles

We classify locally the contact metric ‐spaces whose Boeckx invariant is as tangent hyperquadric bundles of Lorentzian space forms.     

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.     

Vector bundles on curves over and algebraic codes

Let C be a smooth, geometrically connected, projective curve of genus g2 defined over . Here we follow a recent paper of Savin and study the parameters (size, dimension, minimal distance) of algebraic codes constructed using vector bundles E on C. We need to construct vector bundles defined over and with certain numerical invariants (degree, rank h⁰(C,E), order of 1-stability).     

Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces

The aim of this paper is to count subgroups of a given index in the fundamental group of an orientable S ¹-bundle over a compact surface. The number of subgroups of index n turns out to be independent of the orientability of the base surface 𝔣, closed or bordered, and is expressed as a linear combination of the numbers of surface subgroups of indices m = n/l l∣n. For a closed base surface of characteristic χ the respective coefficients are equal to l ^–χm+2 or vanish depending on l,n and the Euler number of the S ¹ -bundle.     

Distributions on almost contact manifolds

It is known that any hypersurface in an almost complex space admits an almost contact manifold [11, 14]. In this article we show that a hyperplane in an almost contact manifold has an almost complex structure. Along with this result, we explain how to determine when an almost contact structure induces a contact structure, followed by examples of a manifold with a closed G₂-structure.     

Vector bundles on a product of real cubic curves

In this paper we give a characterization of then-tuples (C ₁,...,C _n ) of nonsingular projective real cubic curves such that every topological complex vector bundle onC ₁×...×C _n admits an algebraic structure. The results are very explicit and can be expressed in an especially simple form for cubies defined over the rationals.The second author was supported by an NSF grant.     

A lower bound for the number of components of the moduli schemes of stable rank 2 vector bundles on projective 3-folds

Fix a smooth projective 3-fold , , with ample, and . Assume the existence of integers with such that is numerically equivalent to . Let be the moduli scheme of -stable rank 2 vector bundles, , on with and . Let be the number ofits irreducible components. Then .