On the Manifold of Almost Complex Structures

Let (M, g ₀) be a smooth closed Riemannian manifold of even dimension 2_n admitting an almost complex structure. It is shown that the space of all almost complex structures on M determining the same orientation as the one determined by a fixed almost complex structure J ₀ is a smooth locally trivial fiber bundle over the space of almost complex structures orthogonal with respect to g ₀ and determining the same orientation as J ₀.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 66–71.Original Russian Text Copyright © 2005 by N. A. Daurtseva.     

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.     

Curvature vs. Almost Hermitian Structures

A Bochner-type formula for almost Hermitian manifolds is introduced. From this formula, one can find obstructions imposed by the curvature to the existence of certain almost Hermitian structures on compact manifolds.     

Integrability of Analytic Almost Complex Structures on Banach Manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C^*-algebras.Mathematics Subject Classifications (2000): primary 32Q60; secondary 53C15, 58B12.     

Integrability Condition on an Almost Complex Structure and Its Application

We obtain and apply a simple form of the integrability condition of an almost complex structure to the Hopf manifolds.     

On Integrable Conditions of Generalized Almost Complex Structures

Generalized complex geometry is a new kind of geometrical structure which contains complex and symplectic geometry as its special cases. This paper gives the equivalence between the integrable conditions of a generalized almost complex structure in big bracket formalism and those in the general framework.     

Almost Complex Poisson Manifolds

In this paper we consider complex Poisson manifolds and extendthe concept of complex Poisson structure, due to Lichnerowicz to themore general concept of almost complex Poisson structures. Examples ofsuch structures and the associated generalized foliation are given.Moreover, some properties of the complex symplectic structures as wellas of the holomorphic complex Poisson structures are studied.     

殆Hermitian流形中的CR子流形

本文研究了殆Kaehler流形中CR子流形的上同调、CR子波形的分布D及其正交补D⊥的维数大于1的时候,近Kaehler流形中每个全脐非平凡的CR子流形一定是全测地的。最后得到：如果M^～是具有H^～B＞0的近Kaehler流形,那么M^～不允许有混合叶层非凡的CR子流形。     

Complex Product Structures and Affine Foliations

A complex product structure on a manifold is an appropriate combination of a complex structure and a product structure. The existence of such a structure determines many interesting properties of the underlying manifold, notably that the manifold admits a pair of complementary foliations whose leaves carry affine structures. This is due to the existence of a unique torsion-free connection which preserves both the complex and the product structure; this connection is not necessarily flat. We study the existence of complex product structures on tangent bundles of smooth manifolds, and we investigate the structure of manifolds admitting a complex product structure and a compatible hypersymplectic metric, showing that the foliations mentioned earlier are either symplectic or Lagrangian, depending on the symplectic form under consideration.     

Coherent Transforms and Irreducible Representations Corresponding to Complex Structures on a Cylinder and on a Torus

We study a class of algebras with non-Lie commutation relations whose symplectic leaves are surfaces of revolution: a cylinder or a torus. Over each of such surfaces we introduce a family of complex structures and Hilbert spaces of antiholomorphic sections in which the irreducible Hermitian representations of the original algebra are realized. The reproducing kernels of these spaces are expressed in terms of the Riemann theta function and its modifications. They generate quantum Kähler structures on the surface and the corresponding quantum reproducing measures. We construct coherent transforms intertwining abstract representations of an algebra with irreducible representations, and these transforms are also expressed via the theta function.     

On the Constant Type of Almost Hermitian Manifolds

We suggest a most natural generalization of the notion of constant type for nearly Kählerian manifolds introduced by A. Gray to arbitrary almost Hermitian manifolds. We prove that the class of almost Hermitian manifolds of zero constant type coincides with the class of Hermitian manifolds. We show that the class of G ₁-manifolds of zero constant type coincides with the class of 6-dimensional G ₁-manifolds with a non-integrable structure. Finally, we prove that the class of normal G ₂-manifolds of nonzero constant type coincides with the class of 4-dimensional G ₂-manifolds with a nonintegrable structure.     

Note on Left Invariant Almost Cosymplectic Structures

Periodica Mathematica Hungarica -     

关于近Hermitian流形的注记(英文)

We give a necessary and sufficient condition for an almost Hermitian manifold to be a Kahler manifold. By making use of this condition, we give a new proof of Goldberg＇s theorem.     

Harmonic Almost Contact Structures

An almost contact metric structure is parametrized by a section σ of an associated homogeneous fibre bundle, and conditions for σ to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field ξ, and the almost complex structure in the contact subbundle. Several examples are given where the harmonic section equations for σ reduce to those for ξ, regarded as a section of the unit tangent bundle. These include trans-Sasakian structures. On the other hand, there are examples where ξ is harmonic but σ is not a harmonic section. Many examples arise by considering hypersurfaces of almost Hermitian manifolds, with the induced almost contact structure, and comparing the harmonic section equations for both structures.      

Positive Derivations on Archimedean Almost f-Rings

It is shown by P. Colville, G. Davis and K. Keimel that if R is an Archimedean f-ring then a positive group endomorphism D on R is a derivation if and only if the range of D is contained in N(R) and the kernel of D contains R ², where N(R) is the set of all nilpotent elements in R and R ² is the set of all products uv in R. The main objective of this paper is to establish the result corresponding to the Colville–Davis–Keimel theorem in the almost f-ring case. The result obtained in this regard is that if D is a positive derivation in an Archimedean almost f-ring, then the range of D is contained in N(R) and the kernel of D contains R ³, where R ³ is the set of all products uvw in R. Examples are produced showing that, contrary to the f-ring case, the converse is in general false and the third power is the best possible.     

Anti-holomorphic involutions and almost complex structures on four-manifold

Let σ be an anti-holomorphic involution on almost complex manifoldX, the existence of almost complex structure onY=X/σ is discussed. Project supported by the National Natural Science Foundation of China (Grant No. 19771010).     

Maximal Holonomy of Almost Bieberbach Groups for Heis5

Let Heis _2n+1 be the Heisenberg group of dimension 2n + 1 and M an infra-nilmanifold with Heis _2n+1-geometry. The fundamental group of M contains a cocompact lattice of Heis _2n+1 with index bounded above by a universal constant I _n+1, i.e., I _n+1 is the maximal order of the holonomy groups. We prove that I ₃ = 24. As an application we give an estimate for the volumes of finite volume non-compact complex hyperbolic 3-manifolds.     

The Dimension Function of Holomorphic Spaces of a Real Submanifold of an Almost Complex Manifold

Let M be a real submanifold of an almost complex manifold and let be the maximal holomorphic subspace, for each x M. We prove that c:M,c(x)=dim H _x is upper-semicontinuous.     

Almost periodic solutions of neutral impulsive systems with periodic time-dependent perturbed delays

A neutral impulsive system with a small delay of the argument of the derivative and another delay which differs from a constant by a periodic perturbation of a small amplitude is considered. If the corresponding system with constant delay has an isolated ω-periodic solution and the period of the delay is not rationally dependent on ω, then under a nondegeneracy assumption it is proved that in any sufficiently small neighbourhood of this orbit the perturbed system has a unique almost periodic solution.