Obstructions to toric integrable geodesic flows in dimension 3

The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T ^* Q\Q. If the geodesic flow is toric integrable, the cosphere bundle admits the structure of a contact toric manifold. By comparing the Betti numbers of contact toric manifolds and cosphere bundles, we are able to provide necessary conditions for the geodesic flow on a compact, connected 3-dimensional Riemannian manifold to be toric integrable.Mathematics Subject Classifications (2000): primary 53D25; secondary 53D10     

On the geometry of Sasakian-Einstein 5-manifolds

On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [24]. We expand on the recent work of Demailly and Kollár [14] and Johnson and Kollár [20] who give methods for constructing Kähler-Einstein metrics on log del Pezzo surfaces. By a previous result of the first two authors [9], circle V-bundles over log del Pezzo surfaces with Kähler-Einstein metrics have Sasakian-Einstein metrics on the total space of the bundle. Here these simply connected 5-manifolds arise as links of isolated hypersurface singularities which by the well known work of Smale [36] together with [11] must be diffeomorphic to S ⁵ #l(S ² ×S ³ ). More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on S ² ×S ³ and infinite families of such structures on #l(S ² ×S ³ ) with 2≤l≤7. We also discuss the moduli problem for these Sasakian-Einstein structures.     

Five-dimensional K-contact Lie algebras

We introduce a general approach to the study of left-invariant K-contact structures on Lie groups and we obtain a full classification in dimension five. We show that Sasakian structures on five-dimensional Lie algebras with non-trivial center are a relatively rare phenomenon with respect to K-contact structures. We also prove that a five-dimensional solvmanifold with a left-invariant K-contact (not Sasakian) structure is a \mathbb S¹{\mathbb S^1} -bundle over a symplectic solvmanifold. Rigidity results are then obtained for five-dimensional K-contact Lie algebras with trivial center and for K-contact η-Einstein structures. Moreover, five-dimensional Sasakian φ-symmetric Lie algebras are completely classified, and some explicit examples of five-dimensional Sasakian pseudo-metric Lie algebras are provided.     

Almost Contact Lagrangian Submanifolds of Nearly Kaehler 6-Sphere

For a Lagrangian submanifold M of S ⁶ with nearly Kaehler structure, we provide conditions for a canonically induced almost contact metric structure on M by a unit vector field, to be Sasakian. Assuming M contact metric, we show that it is Sasakian if and only if the second fundamental form annihilates the Reeb vector field ξ, furthermore, if the Sasakian submanifold M is parallel along ξ, then it is the totally geodesic 3-sphere. We conclude with a condition that reduces the normal canonical almost contact metric structure on M to Sasakian or cosymplectic structure.     

On Legendre curves in contact 3-manifolds

It is first observed that on a 3-dimensional Sasakian manifold the torsion of a Legendre curve is identically equal to +1. It is then shown that, conversely, if a curve on a Sasakian 3-manifold has constant torsion +1 and satisfies the initial conditions at one point for a Legendre curve, it is a Legendre curve. Furthermore, among contact metric structures, this property is characteristic of Sasakian metrics. For the standard contact structure onR ³ with its standard Sasakian metric the curvature of a Legendre curve is shown to be twice the curvature of its projection to thexy-plane with respect to the Euclidean metric. Thus this metric onR ³ is more natural for the study of Legendre curves than the Euclidean metric.This work was done while the first author was a visiting scholar at Michigan State University.     

g-Natural Contact Metrics on Unit Tangent Sphere Bundles

We construct a three-parameter family of contact metric structures on the unit tangent sphere bundle T ₁ M of a Riemannian manifold M and we study some of their special properties related to the Levi-Civita connection. More precisely, we give the necessary and sufficient conditions for a constructed contact metric structure to be K-contact, Sasakian, to satisfy some variational conditions or to define a strongly pseudo-convex CR-structure. The obtained results generalize classical theorems on the standard contact metric structure of T ₁ M.     

A class of Sasakian 5-manifolds

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H _{2n + 1}. Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H ₅ or a semidirect product ? ? (H ₃ × ?). In particular, the compact quotient is an S ¹-bundle over a four-dimensional Kähler solvmanifold.     

The structure of some classes of <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S ²ⁿ⁺¹ (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S ¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.     

Einstein–Weyl structures on contact metric manifolds

In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W ^± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f η) are presented.      

An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold

Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsváth–Szabó contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston–Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than S³ with negative maximal Thurston–Bennequin invariant. Perhaps more interesting, our invariant provides a criterion for an open book to induce a tight contact structure. A corollary is that if a manifold possesses contact structures with distinct non-vanishing Ozsváth–Szabó invariants, then any fibered knot can realize the classical Eliashberg–Bennequin bound in at most one of these contact structures.     

Sasakian structures on CR-manifolds

A contact manifold M can be defined as a quotient of a symplectic manifold X by a proper, free action of \(\mathbb{R}\), with the symplectic form homogeneous of degree 2. If X is also Kähler, and its metric is homogeneous of degree 2, M is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kähler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold M is CR-diffeomorphic to an S ¹-bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces an embedding of M into an algebraic cone C. We show that this embedding is uniquely defined by the CR-structure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.     

Contact structures on product five-manifolds and fibre sums along circles

Two constructions of contact manifolds are presented: (i) products of S ¹ with manifolds admitting a suitable decomposition into two exact symplectic pieces and (ii) fibre connected sums along isotropic circles. Baykur has found a decomposition as required for (i) for all closed, oriented 4-manifolds. As a corollary, we can show that all closed, oriented 5-manifolds that are Cartesian products of lower-dimensional manifolds carry a contact structure. For symplectic 4-manifolds we exhibit an alternative construction of such a decomposition; this gives us control over the homotopy type of the corresponding contact structure. In particular, we prove that \mathbb CP²×S¹{{\mathbb {CP}}^2\times S^1} admits a contact structure in every homotopy class of almost contact structures. The existence of contact structures is also established for a large class of 5-manifolds with fundamental group \mathbbZ₂{{\mathbb{Z}}_2} .     

New construction of complex manifold via conifold transition

For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth threefolds which is diffeomorphic to connected sums of S3 ×S3 . In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S3 × S3 . This construction is an analogue of that made by Friedman [On threefolds with trivial canonical bundle. In: Complex Geometry and Lie Theory, volume 53 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, 103-134], Lu and Tian [Complex structures on connected sums of S3 × S3 . In: Manifolds and Geometry, Pisa, 1993, 284-293] who used only quintics in P4 .     

<Emphasis Type="Italic">g</Emphasis>-Natural Contact Metrics on Unit Tangent Sphere Bundles

We construct a three-parameter family of contact metric structures on the unit tangent sphere bundle T ₁ M of a Riemannian manifold M and we study some of their special properties related to the Levi-Civita connection. More precisely, we give the necessary and sufficient conditions for a constructed contact metric structure to be K-contact, Sasakian, to satisfy some variational conditions or to define a strongly pseudo-convex CR-structure. The obtained results generalize classical theorems on the standard contact metric structure of T ₁ M. Author supported by funds of the University of Lecce.     

Index theory on locally homogeneous spaces

We describe how the equivariant K homology class of an invariant elliptic operator on a homogeneous space of a linear semisimple Lie group determines the L ²-index of the associated operator on a finite volume locally homogeneous space. The machinery of equivariant K homology and of KK theory can be used to prove theorems about L ²-indices. We give an application motivated by the problem of calculating multiplicities of subrepresentations of quasi-regular representations.Supported by the National Science Foundation under Grant No. DMS-8903472.Supported by the National Science Foundation under Grant No. DMS-8901436.     

Vector bundles on toric varieties

Following Sam Payne?s work, we study the existence problem of nontrivial vector bundles on toric varieties. The first result we prove is that every complete fan admits a nontrivial conewise linear multivalued function. Such functions could potentially be the Chern classes of toric vector bundles. Then we use the results of Cortiñas, Haesemeyer, Walker and Weibel to show that the (non-equivariant) Grothendieck group of the toric 3-fold studied by Payne is large, so the variety has a nontrivial vector bundle. Using the same computation, we show that every toric 3-fold X either has a nontrivial line bundle, or there is a finite surjective toric morphism from Y to X, such that Y has a large Grothendieck group.     

On Positive Sasakian Geometry

A Sasakian structure =(\xi,\eta,\Phi,g) on a manifold Mis called positiveif its basic first Chern class c₁( ) can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This provides us with a new technique for proving the existence of positive Ricci curvature metrics on certain odd dimensional manifolds. As an example we give a completely independent proof of a result of Sha and Yang that for every nonnegative integer kthe 5-manifolds k#(S ²×S ³) admits metrics of positive Ricci curvature.     

L''interpretation de la classe de Maslov dans la K-theorie Hermitienne et dans la theorie relative de Chern-Weil

The purpose of this paper is, first to define the Maslov invariant in the U-theory, the intermediate theory between real K-theory and complex K-theory, whose vanishing is a necessary and sufficient condition for the stable transversality of two Lagrangian sub-bundles of a symplectic fiber bundle. The second purpose is to show, in conformity with Bott periodicity, that Chern, Pontryand Stiefel-Whitney classes are precise enough to play the same role as the Maslov classes when one replaces the base space X of the symplectic bundle, by S ⁷ X ⁺the seventh topological suspension of X ⁺the compactification of X.     

Stability of the Reeb vector field of H-contact manifolds

It is well known that a Hopf vector field on the unit sphere S ²ⁿ⁺¹ is the Reeb vector field of a natural Sasakian structure on S ²ⁿ⁺¹. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction. Supported by funds of the University of Lecce and M.I.U.R.(PRIN).     

Non-zero contact and Sasakian reduction

We complete the reduction of Sasakian manifolds with the non-zero case by showing that Willett's contact reduction is compatible with the Sasakian structure. We then prove the compatibility of the non-zero Sasakian (in particular, contact) reduction with the reduction of the Kähler (in particular, symplectic) cone. We provide examples obtained by toric actions on Sasakian spheres and make some comments concerning the curvature of the quotients.