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.     

Riemannian almost product and para-Hermitian cotangent bundles of general natural lift type

We characterize the almost product and locally product structures of general natural lift type on the cotangent bundle of a Riemannian manifold. We find the conditions under which the cotangent bundle endowed with the constructed almost product (locally product) structure and with a pseudo-Riemannian metric obtained as a general natural lift of the metric from the base manifold, is a Riemannian almost product (locally product) or an (almost) para-Hermitian manifold. Finally, by studying the closedness of the 2-form associated to the obtained (almost) para-Hermitian structure, we characterize the general natural (almost) para-Kählerian structures on the cotangent bundle.     

近Kaehler流形S^3× S^3上的殆切触拉格朗日子流形

对于近Kaehler流形S^3× S^3上的一个拉格朗日子流形M ,给出由M 上的一个单位向量场典范引出的殆切触度量结构是α-Sasakian 的充要条件。当这个殆切触度量结构为切触度量结构时,给出了这个切触度量结构是Sasakian结构的充分必要条件。     

Note on the existence of almost omplex structures on compact manifolds

In the paper we define a multiplicative genus of a compact orientable manifold. We use this genus for the study of the existence of almost complex structures on manifolds. A few applications are given, namely, we prove the nonexistence of an almost complex structure on quaternionic flag manifolds and give a theorem on the existence of an almost complex structure on the product of manifolds.     

Canonical connection on a class of Reimannian almost product manifolds

The canonical connection on a Reimannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Reimannian almost product manifolds with non-integrable almost product structure.     

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.     

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.      

Oka's principle for holomorphic submersions with sprays

We classify, up to a local isometry, all non-K?hler almost K?hler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost K?hler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure. Received: 1 October 2001 / Published online: 17 June 2002     

The sharp Jackson inequality in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> on the segment [−1, 1] with the power weight

It is proved that there exists an equivariant almost complex structure on any quasitoric manifold that admits a positive omniorientation. This gives an answer to the question raised by M. Davis and T. Januszkiewicz: Find a criterion for the existence of an equivariant almost complex structure on a quasitoric manifold in terms of its characteristic function.     

Maximally homogeneous para-CR manifolds

We define the notion of a (weak) almost para-CR structure on a manifold M as a distribution HM ⊂ TM together with a field K ∈ Γ(End(HM)) of involutive endomorphisms of HM. If K satisfies integrability conditions, then (HM, K) is called a (weak) para-CR structure. Under some regularity conditions, an almost para-CR structure can be identified with a Tanaka structure. The notion of maximally homogeneous almost para-CR structure of a semisimple type is defined. A classification of such maximally homogeneous almost para-CR structures is given in terms of appropriate gradations of real semisimple Lie algebras. All such maximally homogeneous structures of depth two (which correspond to depth two gradations) are listed and the integrability conditions are verified.Mathematics Subject Classifications (1991): 53C15, 53D99, 58A14     

Homogeneous almost complex structures in dimension 6 with semi-simple isotropy

We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homogeneous types or a left-invariant structure on a Lie group. For integrable invariant almost complex structures we classify all compatible invariant Hermitian structures on these homogeneous manifolds, indicate their integrability properties (Kähler, SNK, SKT) and mark the other interesting geometric properties (including the Gray-Hervella type).     

Invariants of conformal transformations of almost contact metric structures

The so-called structure tensors of almost contact metric structures, which play a key role in the geometry of almost contact metric structures, are explicitly calculated. The transformations of these tensors under conformal transformations of almost contact metric structures are described. The results obtained are used to study the behavior of the most interesting classes of almost contact structures under conformal transformations.     

A Note on Donaldson's “Tamed to Compatible” Question

Recently, Tedi Draghici and Weiyi Zhang studied Donaldson's "tamed to compatible" question(Draghici T, Zhang W. A note on exact forms on almost complex manifolds. arXiv: 1111. 7287v1 [math. SG]. Submitted on 30 Nov. 2011).That is, for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form, is there a symplectic form compatible with this almost complex structure? They got several equivalent forms of this problem by studying the space of exact forms on such a manifold. With these equivalent forms, they proved a result which can be thought as a further partial answer to Donaldson's question in dimension 4. In this note, we give another simpler proof of their result.     

On generalized quasi-Sasaki manifolds

We study 5-dimensional Riemannian manifolds that admit an almost contact metric structure. In particular, we generalize the class of quasi-Sasaki manifolds and characterize these structures by their intrinsic torsion. Among other things, we see that these manifolds admit a unique metric connection that is compatible with the underlying almost contact metric structure. Finally, we construct a family of examples that are not quasi-Sasaki.     

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.     

A Note on Donaldson＇s ＂Tamed to Compatible＂, Question

Recently, Tedi Draghici and Weiyi Zhang studied Donaldson＇s ＂tamed to compatible＂ question （Draghici T, Zhang W. A note on exact forms on almost complex manifolds, arXiv： 1111. 7287vl [math. SC]. Submitted on 30 Nov. 2011）. That is, for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form, is there a symplectic form compatible with this almost complex structure？ They got several equivalent forms of this problem by studying the space of exact forms on such a manifold. With these equivalent forms, they proved a result which can be thought as a further partial answer to Donaldson＇s question in dimension 4. In this note, we give another simpler proof of their result.     

Stability and extensibility results for abstract skew-product semiflows

In this paper we present new stability and extensibility results for skew-product semiflows with a minimal base flow. In particular, we describe the structure of uniformly stable and uniformly asymptotically stable sets admitting backwards orbits and the structure of omega-limit sets. As an application, the occurrence of almost periodic and almost automorphic dynamics for monotone non-autonomous infinite delay functional differential equations is analyzed.     

Relative nullity distributions, an affine immersion from an almost product manifold and a para-pluriharmonic isometric immersion

We give a characterization for an affine immersion from an almost product manifold the almost product structure of which is adjoint or skew-adjoint with respect to its affine fundamental form by the relative nullity distribution.     

Almost Hermitian structures induced from a Kähler structure which has constant holomorphic sectional curvature

We obtain a non-Kähler almost Hermitian manifold of constant holomorphic sectional curvature by changing the almost complex structure in a Kähler manifold of constant holomorphic sectional curvature.

     

Local Riemann-Roch theorem for almost Hermitian manifolds

In this paper, we give a local Riemann-Roch theorem for certain almost Hermitian manifolds by a canonical connection and the theory of almost Hermitian structure of null eccentricity. In particular, we obtain a local Riemann-Roch theorem for all closed almost complex manifolds of dimension 4. Also, by using the heat equation method, we give an analytic approach to a local index theorem for generalized Dirac operators.