Discussion on the Complex Structure of Nilpotent Lie Groups Gk

Consider the real, simply-connected, connected, s-step nilpotent Lie group G endowed with a left-invariant, integrable almost complex structure J, which is nilpotent. Consider the simply-connected, connected nilpotent Lie group Gk, defined by the nilpotent Lie algebra g/ak, where g is the Lie algebra of G, and ak is an ideal of g. Then, J gives rise to an almost complex structure Jk on Gk. The main conclusion obtained is as follows: if the almost complex structure J of a nilpotent Lie group G is nilpotent, then J can give rise to a left-invariant integrable almost complex structure Jk on the nilpotent Lie group Gk, and Jk is also nilpotent.

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.

ISOPARAMETRIC HYPERSURFACES AND COMPLEX STRUCTURES

The main purpose of this note is to construct almost complex or complex structures on certain isoparametric hypersurfaces in unit spheres.As a consequence,complex structures on S^(1)×S^(7)×S^(6),and on S^(10)×S^(3)×S(2)with vanishing first Chern class,are built.

Extension formulae on almost complex manifolds

We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae.As applications,we study(n,0)-forms,the(n,0)-Dolbeault cohomology group and(n,q)-forms on almost complex manifolds.

On Tamed Almost Complex Four‑Manifolds

This paper proves that on any tamed closed almost complex four-manifold(M,J)whose dimension of J-anti-invariant cohomology is equal to the self-dual second Betti number minus one,there exists a new symplectic form compatible with the given almost complex structure J.In particular,if the self-dual second Betti number is one,we give an affirmative answer to a question of Donaldson for tamed closed almost complex four-manifolds.Our approach is along the lines used by Buchdahl to give a unified proof of the Kodaira conjecture.

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.

On the vortex filament in 3-spaces and its generalizations

In this article,we devote to a mathematical survey on the theory of the vortex filament in 3-dimensional spaces and its generalizations.We shall present some effective geometric tools applied in the study,such as the Schrodinger flow,the geometric Korteweg-de Vries(KdV)flow and the generalized bi-Schrodinger flow,as well as the complex and para-complex structures.It should be mentioned that the investigation in the imaginary part of the octonions looks very fascinating,since it relates to almost complex structures and the G2 structure on S^(6).As a new result in this survey,we describe the equation of generalized bi-Schr?dinger flows from R1 into a Riemannian surface.