On the Proper Homotopy Invariance of the Tucker Property

A non-compact polyhedron P is Tucker if, for any compact subset K begong to P, the fundamental group π1 （P - K） is finitely generated. The main result of this note is that a manifold which is proper homotopy equivalent to a Tucker polyhedron is Tucker. We use Poenaru＇s theory of the equivalence relations forced by the singularities of a non-degenerate simplicial map.     

THE NEW INTEGRABLE SYSTEM OF THE COMPLEX FORM AND THECOUPLED MKDV HIERARCHY

In this paper, the new coupled MKdV hierarchy and their Lsx pairs are oinained,Through introducing a suitable complex form of symplectic structure [5,8], a new integraHe sys-tem of the complex form in the Liouville sense is generated. Moreover, the representations of the solution for the coupled MKdV hierarchy are given by the invohifive solutions of the commutable .     

Square-free integers as ideal norms

Elaborating on Katzel＇s result, Zhai obtained a sharp short interval result for the number of ways of expressing a square-free integer as sums of two squares. In this note we view Zhai＇s decom- position as one by degree 1 prime ideals in the Gaussian field and extend Zhai＇s approach to a wider range of quadratic fields etc.     

Qualitative Analysis of Bifurcating Solutions in the Lengyel-Epstein Model

One of the most fundamental problems in theoretical biology is to explain the mechanisms by which patterns and forms are created in the＇living world. In his seminal paper ＂The Chemical Basis of Morphogenesis＂, Turing showed that a system of coupled reaction-diffusion equations can be used to describe patterns and forms in biological systems. However, the first experimental evidence to the Turing patterns was observed by De Kepper and her associates（1990） on the CIMA reaction in an open unstirred reactor, almost 40 years after Turing＇s prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. The Lengyel-Epstein model is in the form as follows     

$(\omega_1)$性质与单值延拓性质

In this note we study the property（ω1）,a variant of Weyl＇s theorem by means of the single valued extension property,and establish for a bounded linear operator defined on a Banach space the necessary and sufficient condition for which property（ω1） holds.As a consequence of the main result,the stability of property（ω1） is discussed.     

Classification of Lagrangian Surfaces of Curvature e in Non-flat Lorentzian Complex Space Form M12（4ε）

It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form M12（4ε） of constant holomorphic sectional curvature 4s is of constant curvature 6. A natural question is ＂Besides totally geodesic ones how many Lagrangian surfaces of constant curvature εin M12（46） are there？＂ In an earlier paper an answer to this question was obtained for the case e = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ε≠0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ε in M12（4ε） with ε ≠ 0. Conversely, every Lagrangian surface of curvature ε≠0 in M12（4ε） is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.     

Problems of Lifts in Symplectic Geometry

Let(M,ω)be a symplectic manifold.In this paper,the authors consider the notions of musical(bemolle and diesis)isomorphisms ω~b:T M→T~*M and ω~?:T~*M→TM between tangent and cotangent bundles.The authors prove that the complete lifts of symplectic vector field to tangent and cotangent bundles is ω~b-related.As consequence of analyze of connections between the complete lift ~cω_(T M )of symplectic 2-form ω to tangent bundle and the natural symplectic 2-form dp on cotangent bundle,the authors proved that dp is a pullback o f~cω_(TM)by ω~?.Also,the authors investigate the complete lift ~cφ_T~*_M )of almost complex structure φ to cotangent bundle and prove that it is a transform by ω~?of complete lift~cφ_(T M )to tangent bundle if the triple(M,ω,φ)is an almost holomorphic A-manifold.The transform of complete lifts of vector-valued 2-form is also studied.     

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.     

性质$(\omega)$的注解

In this note we study the property （ω）, a variant of Weyl＇s theorem introduced by Rakocevic, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property （ω） and approximate Weyl＇s theorem hold. As a consequence of the main result, we study the property （ω） and approximate Weyl＇s theorem for a class of operators which we call the λ-weak-H（p） operators.     

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.     

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.     

Note on a theorem of Bangert

We generalize Bangert’s non-hyperbolicity result for uniformly tamed almost complex structures on standard symplectic ℝ²ⁿ to asymptotically standard symplectic manifolds.     

From Taubes currents to almost Kähler forms

The Taubes current is introduced by Taubes in (J Symplectic Geom 9:161–250, 2011) as an intermediate step to construct almost Kähler forms. In this paper, we prove that an almost complex structures being almost Kähler structure in dimension four is the equivalent of the existence of Taubes currents. Precisely, we show that Taubes currents could be regularized to almost Kähler forms up to small perturbations of cohomology classes on any 4-dimensional almost complex manifold $(M, J)$ . A similar result is established for higher dimensions under the assumption of almost Kähler. An application to Donaldson’s “tamed to compatible” question is provided.     

Locally Lagrange-Symplectic Manifolds

The symplectic forms which have a compatible tangent structure locally look like the symplectic form of a nondegenerate Lagrangian function. For this reason, the corresponding manifold is said to be locally Lagrange-symplectic. In the paper, we give some elementary properties of these manifolds, and discuss their symplectic reduction.     

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.     

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.     

Compatible Almost Complex Structures on Quaternion Kähler Manifolds

Let (M⁴ⁿ,g,Q) be a quaternion Kähler manifold with reduced scalar curvature = K/4n(n + 2). Suppose J is an almost complex structure which is compatible with the quaternionic structure Q and let = – F J be the Lee form of J. We prove the following local results: (1) if J is conformally symplectic, then it is parallel and = 0; (2) if J is cosymplectic, then 0 with equality if and only if J is parallel; (3) if J is integrable, then d is Q-Hermitian and harmonic; and (4) any closed self-dual 2-form = f(g J) ² ₊ = g Q ² is parallel. In Section 5, extending previous results of Salamon [24], we describe a correspondence among conformally balanced J, Killing vector fields X and self-dual 2-forms satisfying the twistor equation.When M⁴ⁿ is compact our main global results are the following: (1) if > 0, then there exists no compatible almost complex structure J; (2) if the first Chern class c₁(T^(1,0) _J M) = 0, then = 0; (3) if = 0 a compatible complex structure J is parallel; and (4) if 0, then no compatible complex structure J exists. The last two results have been proved in [23] by twistor methods.     

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.