Fano contact manifolds and nilpotent orbits

A contact structure on a complex manifold M is a corank 1 subbundle F of T_M such that the bilinear form on F with values in the quotient line bundle L = T_M/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.?If is a simple Lie algebra, the unique closed orbit in (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.?In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M P(H ₀(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant. Received: July 28, 1997     

On the existence of almost contact structure and the contact magnetic field

We give a simple proof of the existence of an almost contact metric structure on any orientable 3-dimensional Riemannian manifold (M ³, g) with the prescribed metric g as the adapted metric of the almost contact metric structure. By using the key formula for the structure tensor obtained in the proof this theorem, we give an application which allows us to completely determine the magnetic flow of the contact magnetic field in any 3-dimensional Sasakian manifold.     

Sur l'existence du nombre de Lelong d'un courant positif fermé défini sur une variété presque complexe

Let (M,J) be an almost complex manifold. By using local coordinate system adapted to the structure J, we prove that every closed positive current on M possesses a Lelong number at any point. In case the manifold is equipped with an integrable complex structure, this Lelong number coincides with the usual Lelong number of a closed positive current.     

Weak stability of almost regular contact foliations

We prove that on a compact manifold, a contact foliation obtained by a smallC ¹ perturbation of an almost regular contact flow has at least two closed characteristics. This solves the Weinstein conjecture for contact forms which areC ¹-close to almost regular contact forms.Supported in part by NSF Grant DMS 90-01861     

Canonical Foliations of Certain Classes of Almost Contact Metric Structures

The purpose of this paper is to study the canonical foliations of an almost cosymplectic or almost Kenmotsu manifold M in a unified way. We prove that the canonical foliation F defined by the contact distribution is Riemannian and tangentially almost Kahler of codimension 1 and that F is tangentially Kahler if the manifold M is normal. Furthermore, we show that a semi-invariant submanifold N of such a manifold M admits a canonical foliation FN which is defined by the antiinvariant distribution and a canonical cohomology class c（N） generated by a transversal volume form for FN. In addition, we investigate the conditions when the even-dimensional cohomology classes of N are non-trivial. Finally, we compute the Godbillon Vey class for FN.     

Foliations and Complemented Framed Structures on an Almost Contact Metric Manifold

On an odd dimensional manifold, we define a structure which generalizes several known structures on almost contact manifolds, namely Sasakian, trans-Sasakian, quasi-Sasakian, Kenmotsu and cosymplectic structures. This structure, hereinafter called a generalized quasi-Sasakian, shortly G.Q.S. structure, is defined on an almost contact metric manifold and satisfies an additional condition. Then we consider a distribution D₁{\mathcal{D}_{1}} wich allows a suitable decomposition of the tangent bundle of a G.Q.S. manifold. Necessary and sufficient conditions for the normality of the complemented framed structure on the distribution D₁{\mathcal{D}_{1}} defined on a G.Q.S manifold are studied. The existence of the foliation on G.Q.S. manifolds and of bundle-like metrics are also proven. It is shown that under certain circumstances a new foliation arises and its properties are investigated. Some examples illustrating these results are given in the final part of this paper.     

On doubly warped product Finsler manifolds

In this paper, we introduce horizontal and vertical warped product Finsler manifolds. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemannian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. We obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the tangent bundle of a doubly warped product Finsler manifold.     

A new class of almost complex structures

The purpose of this paper is to introduce a new class of almost complex structures J on a Riemannian manifold M by using a certain identity for the relationship between the tensor F _i ^j of J and the Riemann curvature tensor R _hijk of M. This class contains the Kählerian structures, and its relationship with some known classes of almost Hermitian structures defined by similar identities is discussed. For convenience we call each structure of this new class an almost C-structure, and a manifold with an almost C-structure an almost C-manifold. We obtain an analogue of F. Schur's theorem concerning the holomorphic sectional curvature of an almost Hermitian C-manifold, and some sufficient conditions for an almost Hermitian C-manifold to be Kählerian. We show that these results are also true for a manifold with a complex structure.     

Certain CR submanifolds of maximal CR dimension of complex space forms

We study m-dimensional real submanifolds with (m−1)-dimensional maximal holomorphic tangent subspace in complex space forms. On such a manifold there exists an almost contact structure which is naturally induced from the ambient space and in this paper we study the anti-commutative condition of the almost contact structure and the second fundamental form of these submanifolds and we characterize certain model spaces in complex space forms.     

关于切丛和单位切球丛的度量的一个注记

本文在黎曼流形$(M,g)$的切丛$TM$ 上研究与参考文献[10]中平行的一类度量$G$以及相容的近复结构$J$.证明了切丛$TM$关于这些度量和相应的近复结构是局部共形近K\"{a}hler流形,并且把这些结构限制在单位切球丛上得到了切触度量结构的新例子.     

On submanifolds of Lxf F satisfying Chen's basic equality

It is well known that the warped product Lx_f F of a line L and a Kaehler manifold F is an almost contact Riemannian manifold which is characterized by some tensor equations appeared in (1.7) and (1.8). In this paper we determine submanifolds of Lx_f F which are tangent to the structure vector field and satisfy Chen's basic equality. Also, we investigate tubular hypersurfaces of Lx_f CE ^m which satisfy Chen's basic equality where CE ^m is a complex Euclidean m-space. This revised version was published online in June 2006 with corrections to the Cover Date.     

Contact Pseudo-Metric Manifolds of Constant Curvature and CR Geometry

In this paper, we show that if an integrable contact pseudo-metric manifold of dimension 2n + 1, n ≥ 2, has constant sectional curvature \({\kappa}\) , then the structure is Sasakian and \({\kappa=\varepsilon=g(\xi,\xi)}\) , where \({\xi}\) is the Reeb vector field. We note that the notion of contact pseudo-metric structure is equivalent to the notion of non-degenerate almost CR manifold, then an equivalent statement of this result holds in terms of CR geometry. Moreover, we study the pseudohermitian torsion \({\tau}\) of a non-degenerate almost CR manifold.     

Almost contact Kähler manifolds of constant holomorphic sectional curvature

We introduce the notion of an almost contact Kähler structure. We also define the holomorphic sectional curvature of the distribution of an almost contact Kähler structure with respect to an interior metric connection and establish relations between the φ-sectional curvature of an almost contact Kähler manifold and the holomorphic sectional curvature of the distribution of an almost contact Kähler structure.     

Five-Dimensional Contact SU(2)- and SO(3)-Manifolds and Dehn Twists

In the first part of this paper the five-dimensional contact SO(3)-manifolds are classified up to equivariant coorientation preserving contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists. In an appendix we also describe the classification of five-dimensional contact SU(2)-manifolds.     

The Yamabe problem on quaternionic contact manifolds

By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(ℍ ⁿ ). Mathematics Subject Classification (2000) 53C17, 53D10, 35J70     

Sur l'existence du nombre de Lelong d'un courant positif psh défini sur une variété presque complexe

The goal of this Note is to extend to an almost complex manifold the existence of the Lelong number of a positive plurisubharmonic (psh) current. In this way, we generalize results of Lelong and Skoda established in the case of an integrable complex structure, and of Haggui in the non-integrable case, but only for a closed positive current. The main point is to establish a Lelong–Jensen formula for a positive psh current defined on an almost complex manifold, which generalizes a formula proved by Demailly when the structure is integrable. To cite this article: F. Elkhadhra, S.K. Mimouni, C. R. Acad. Sci. Paris, Ser. I 344 (2007).     

<Emphasis Type="Italic">N</Emphasis>-extended symplectic connections in almost contact metric spaces

On a manifold with an almost contact metric structure we introduce the notions of intrinsic connection, N-extended connection and N-connection. It is shown that the Tanaka–Webster and Schouten–van Kampen connections are special cases of N-connection. We define new classes of N-connections, namely, the Vagner connection and canonical metric N-connection. We also define N-extended symplectic connection. It is proved that the N-extended symplectic connection exists on any manifold with a contact metric structure.     

Curvature identities for principle <Emphasis Type="Italic">T</Emphasis><Superscript>1</Superscript>-bundles over almost Hermitian manifolds

We show that the identities R ₁, R ₂ and R ₃ for an almost Hermitian structure S on the base of the canonical principal T ¹-bundle are equivalent to their contact analogs for the induced almost contact metric structure S ^# on the total space of this bundle. We prove that the canonical connection of the canonical principal T ¹-bundle over a Hermitian or a quasi-Kähler manifold of class R ₃ is normal. We also prove that that the canonical connection of the canonical principal T ¹-bundle over a Vaisman-Gray manifold M of class R ₃ is normal if and only if the Lee vector of M belongs to the center of the adjoint K-algebra.     

Anti-Kählerian manifolds

An anti-Kählerian manifold is a complex manifold with an anti-Hermitian metric and a parallel almost complex structure. It is shown that a metric on such a manifold must be the real part of a holomorphic metric. It is proved that all odd Chern numbers of an anti-Kählerian manifold vanish and that complex parallelisable manifolds (in particular the factor space G/D of a complex Lie group G over the discrete subgroup D ) are anti-Kählerian manifolds. A method of generating new solutions of Einstein equations by using the theory of anti-Kählerian manifolds is presented.     

Some Remarks on Almost and Stable Almost Complex Manifolds

In the first part we give necessary and sufficient conditions for the existence of a stable almost complex structure on a 10-manifold M with H₁(M;?) = 0 and no 2-torsion in H₁(M;?) for i = 2,3. Using the Classification Theorem of Donaldson we give a reformulation of the conditions for a 4-manifold to be almost complex in terms of Betti numbers and the dimension of the ±-eigenspaces of the intersection form. In the second part we give general conditions for an almost complex manifold to admit infinitely many almost complex structures and apply these to symplectic manifolds, to homogeneous spaces and to complete intersections.