On Almost Hermitian 4-manifolds with J-invariant Ricci Tensor

We prove that every almost Hermitian 4-manifold with J-invariant Ricci tensor which is conformally flat or has harmonic curvature is either a space of constant curvature or a Kähler manifold. We also obtain analogous results on almost Kähler 4-manifolds.     

Compact Almost Kähler Manifolds with Divergence-Free Weyl Conformal Tensor

We prove that a compact almost Kähler manifold satisfying that a certain part of thedivergence W of the Weyl conformal tensor W vanishes isKähler.     

On the Twistor Spaces of Almost Hermitian Manifolds

We study Grassmannian bundles G_k(M) of analytical 2k-planes over an almost Hermitian manifold M²ⁿ, from the point of view of the generalized twistor spaces of [13], and with the method of the moving frame [9]. G₁(M⁴) is the classical twistor space. We find four distinguished almost Hermitian structures, one of them being that of [13], and discuss their integrability and Kählerianity. For n=2, we compute the corresponding Hermitian connections, and derive consequences about the corresponding first Chern classes.     

Dirichlet problem for Hermitian Yang-Mills-Higgs equations over Hermitian manifolds

In this paper, we investigate the Dirichlet problem for one type of vortex equations, which generalize the well-known Hermitian Yang-Mills equations, over general Hermitian manifolds.     

Holomorphic Torus Actions on Compact Locally Conformal Kähler Manifolds

Given a torus action (T ², M) on a smooth manifold, the orbit map ev _x(t)=t·xfor each xMinduces a homomorphism ev _*: ²H ₁(M;). The action is said to be Rank-kif the image of ev _*has rank k(2) for each point of M. In particular, if ev _*is a monomorphism, then the action is called homologically injective. It is known that a holomorphic complex torus action on a compact Kähler manifold is homologically injective. We study holomorphic complex torus actions on compact non-Kähler Hermitian manifolds. A Hermitian manifold is said to be a locally conformal Kähler if a lift of the metric to the universal covering space is conformal to a Kähler metric. We shall prove that a holomorphic conformal complex torus action on a compact locally conformal Kähler manifold Mis Rank-1 provided that Mhas no Kähler structure.     

Almost Complex Structures Which Are Compatible with Kähler or Symplectic Structures

In this note we prove that half of all homotopy classes of almost complex structures on M is not compatible with any symplectic structure for a certain class of oriented compact 4-manifolds M. In particular, half of all homotopy classes of almost complex structures on an oriented 4-manifold is not compatible to any Kähler structure.     

Conformally Einstein products and nearly Kähler manifolds

In the first part of this note we study compact Riemannian manifolds (M, g) whose Riemannian product with is conformally Einstein. We then consider 6-dimensional almost Hermitian manifolds of type W ₁ + W ₄ in the Gray–Hervella classification admitting a parallel vector field and show that (under some mild assumption) they are obtained as Riemannian cylinders over compact Sasaki–Einstein 5-dimensional manifolds.      

Four-Dimensional Almost Kähler Einstein and *-Einstein Manifolds

In the framework of studying the integrability of almost Kähler manifolds, we prove that a four-dimensional almost Kähler Einstein and -Einstein manifold is a Kähler manifold. Further, we estimate the *-scalar curvature of a four-dimensional compact almost Kähler Einstein and weakly *-Einstein manifold with negative scalar curvature.     

Riemannian manifolds with structure groupG 2

Summary Riemannian manifolds with structure group G ₂ are 7-dimensional and have a distinguished 3-form. In this paper such manifolds are treated as analogues of almost Hermitian manifolds. Thus S ⁷ has structure group G ₂ just as S ⁶ is an almost Hermitian manifold. We study the covariant derivative of the fundamental 3-form as was done in [GH]for almost Hermitian manifolds.     

On the Kenmotsu Hypersurfaces of Special Hermitian Manifolds

We consider almost contact metric hypersurfaces of almost Hermitian manifolds of class W₃ (in the Gray–Hervella terminology). We establish a criterion for minimality of such hypersurfaces in the case when the contact metric structure is cosymplectic.     

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.     

On balanced Hermitian structures on Lie groups

We present several methods for the construction of balanced Hermitian structures on Lie groups. In our methods a partial differential equation is involved so that the resulting structures are in general non homogeneous. In particular, we prove that for 3-step nilpotent Lie groups G of dimension 6, any left-invariant complex structure on G admits a balanced Hermitian metric. Starting from normal almost contact structures, we construct balanced metrics on 6-dimensional manifolds, generalizing warped products. Finally, explicit balanced Hermitian structures are also given on solvable Lie groups defined as semidirect products ${\mathbb{R}^k \ltimes \mathbb{R}^{2n-k}}$ .     

Extremal Kähler metrics on projective bundles over a curve

Let M=P(E) be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle E→Σ over a compact complex curve Σ of genus ?2. Building on ideas of Fujiki (1992) [27], we prove that M admits a Kähler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of E as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in Kähler classes sufficiently far from the boundary of the Kähler cone. The methods used to prove the above results apply to a wider class of manifolds, called rigid toric bundles over a semisimple base, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.     

Generalized Elliptic Genus and Cobordism Class of Nonspin Real Grassmannians

We prove that Witten's generalized elliptic genusis rigid under certain group actions and constant(identically zero) on the real Grassmanniansr₄^2m+5, m 1, for appropriate vectorbundles.We also prove that all the Pontryagin numbers of theseGrassmannians are zero, i.e. they have cobordism class zero,which implies the vanishing of all genera.     

Almost hermitian manifolds and Osserman condition

Let(M, g, J) be an almost Hermitian manifold. In this paper we study holomorphically nonnegatively(δ,Δ)-pinched almost Hermitian manifolds. In [3] it was shown that for such Kahler manifolds a plane with maximal sectional curvature has to be a holomorphic plane(J-invariant). Here we generalize this result to arbitrary almost Hermitian manifolds with respect to the holomorphic curvature tensorH R and toRK-manifolds of a constant type λ(p). In the proof some estimates of the sectional curvature are established. The results obtained are used to characterize almost Hermitian manifolds of constant holomorphic sectional curvature (with respect to holomorphic and Riemannian curvature tensor) in terms of the eigenvalues of the Jacobi-type operators, i.e. to establish partial cases of the Osserman conjecture. Some examples are studied. The first author is partially supported by SFS, Project #04M03.     

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} .     

The Totally Geodesic Coisotropic Submanifolds in Kähler Manifolds

In this paper, we consider the coisotropic submanifolds in a Kähler manifold of nonnegative holomorphic curvature. We prove an intersection theorem for compact totally geodesic coisotropic submanifolds and discuss some topological obstructions for the existence of such submanifolds. Our results apply to Lagrangian submanifolds and real hypersurfaces since the class of coisotropic submanifolds includes them. As an application, we give a fixed-point theorem for compact Kähler manifolds with positive holomorphic curvature. Also, our results can be further extended to nearly Kähler manifolds.     

A Note on the Moment Map on Compact Kähler Manifolds

We consider compact Kähler manifolds acted on effectively by a connected compact Lie group K of isometries in a Hamiltonian fashion. We prove that the squared moment map ||||² is constant if and only if K is semisimple and the manifold is biholomorphically and K-equivariantly isometric to a product of a flag manifold and a compact Kähler manifold which is acted on trivially by K.     

Hyper-Hermitian Quaternionic Kähler Manifolds

We call a quaternionic Kähler manifold with nonzero scalar curvature, whosequaternionic structure is trivialized by a hypercomplex structure, ahyper-Hermitian quaternionic Kähler manifold. We prove that every locallysymmetric hyper-Hermitian quaternionic Kähler manifold is locally isometricto the quaternionic projective space or to the quaternionic hyperbolic space.We describe locally the hyper-Hermitian quaternionic Kähler manifolds withclosed Lee form and show that the only complete simply connected suchmanifold is the quaternionic hyperbolic space.     

On the geometry of principalT 1-bundles over Hodge manifolds

We study Sasakian structures induced in principalT ¹-bundles over Kähler manifolds. A natural model of a Sasakian manifold of constant -holomorphic sectional curvature –3 is constructed.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 824–829, December, 1998.The author is greatly indebted to Professor V. F. Kirichenko for setting the problem, as well as for interest and help during the research.