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...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.展开更多
Over an oriented even dimensional Riemannian manifold(M 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description ...Over an oriented even dimensional Riemannian manifold(M 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description of the twistor bundle Гm = SO(2m)/U(m)? J +(@#@ M,ds2 ) →M. The integrability on an almost complex structureJ compatible with the metric and the orientation, is shown to be equivalent to the fact that the corresponding cross section of the twistor bundle is holomorphic with respect toJ and the canonical almost complex structureJ 1 onJ +(M,ds2 ), by using moving frame theory. Moreover, for various metrics and a fixed orientation onM, a canonical bundle isomorphism is established. As a consequence, we generalize a celebrated theorem of LeBrun.展开更多
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...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.展开更多
Ⅰ. INTRODUCTION Let M be a 2n-dimensional manifold, which will always be assumed to be smooth, closed, connected and oriented. M is said to have an almost complex-structure if there exists a complex n-plane bundle ω...Ⅰ. INTRODUCTION Let M be a 2n-dimensional manifold, which will always be assumed to be smooth, closed, connected and oriented. M is said to have an almost complex-structure if there exists a complex n-plane bundle ω over M whose underlying real 2n-plane bundle is isomorphic to τM the tangent bundle of M.展开更多
The existence or nonexistenoe of weakly almost complex structures and almost complex structures on Dold manifolds are studied, and the problems are partially solved.
1. Let S<sup>4</sup> be a four-sphere and let G<sub>2</sub>（TS<sup>4</sup>） be the Grassmann bundle on S<sup>4</sup> with natural Riemann metric and almost complex str...1. Let S<sup>4</sup> be a four-sphere and let G<sub>2</sub>（TS<sup>4</sup>） be the Grassmann bundle on S<sup>4</sup> with natural Riemann metric and almost complex structure. G<sub>2</sub>（TS<sup>4</sup>） is called （1, 2）-symplectic if the （1, 2）part of dk is zero where k is the K（?）hler form of G<sub>2</sub>（TS<sup>4</sup>）. In this note, we prove the following theorem:展开更多
Let σ be an anti-holomorphic involution on an almost complex four manifold X,a necessary and sufficient condition is given to determine weather X/σ admits an almost complex structure.
Let X be a finite CW complex, and let ξ be a real vector bundle over X. We say that ξ has a complex structure if it is isomorphic to the real bundle r(ω)underlying some complex vector bundle ω over X. Let M be a c...Let X be a finite CW complex, and let ξ be a real vector bundle over X. We say that ξ has a complex structure if it is isomorphic to the real bundle r(ω)underlying some complex vector bundle ω over X. Let M be a closed connected smooth manifold. We say that M has an almost structure if its tangent bundle has a complex structure.展开更多
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 ...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.展开更多
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 ...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.展开更多
文摘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.
基金This work was supported partially by the National Natural Science Foundation of China(Grant No.10131020)Outstanding Youth Foundation of China No.19925103 and No.10229101the“973”.
文摘Over an oriented even dimensional Riemannian manifold(M 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description of the twistor bundle Гm = SO(2m)/U(m)? J +(@#@ M,ds2 ) →M. The integrability on an almost complex structureJ compatible with the metric and the orientation, is shown to be equivalent to the fact that the corresponding cross section of the twistor bundle is holomorphic with respect toJ and the canonical almost complex structureJ 1 onJ +(M,ds2 ), by using moving frame theory. Moreover, for various metrics and a fixed orientation onM, a canonical bundle isomorphism is established. As a consequence, we generalize a celebrated theorem of LeBrun.
基金The project is partially supported by the NSFC(11871282,11931007)BNSF(Z190003)Nankai Zhide Foundation.
文摘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.
文摘Ⅰ. INTRODUCTION Let M be a 2n-dimensional manifold, which will always be assumed to be smooth, closed, connected and oriented. M is said to have an almost complex-structure if there exists a complex n-plane bundle ω over M whose underlying real 2n-plane bundle is isomorphic to τM the tangent bundle of M.
基金National Natural Science Foundation of Chinathe State Education Commission Foundation of Chinathe Foundation of the Academy of Sciences.
文摘The existence or nonexistenoe of weakly almost complex structures and almost complex structures on Dold manifolds are studied, and the problems are partially solved.
文摘1. Let S<sup>4</sup> be a four-sphere and let G<sub>2</sub>（TS<sup>4</sup>） be the Grassmann bundle on S<sup>4</sup> with natural Riemann metric and almost complex structure. G<sub>2</sub>（TS<sup>4</sup>） is called （1, 2）-symplectic if the （1, 2）part of dk is zero where k is the K（?）hler form of G<sub>2</sub>（TS<sup>4</sup>）. In this note, we prove the following theorem:
基金supported by the National Natural Science Foundation of China(Grant No.10371008).
文摘Let σ be an anti-holomorphic involution on an almost complex four manifold X,a necessary and sufficient condition is given to determine weather X/σ admits an almost complex structure.
文摘Let X be a finite CW complex, and let ξ be a real vector bundle over X. We say that ξ has a complex structure if it is isomorphic to the real bundle r(ω)underlying some complex vector bundle ω over X. Let M be a closed connected smooth manifold. We say that M has an almost structure if its tangent bundle has a complex structure.
基金The first author was supported by National Natural Science Foundation of China(Grant Nos.11531012,11926307 and 12071080).
文摘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.
基金The NSF(11071208 and 11126046)of Chinathe Postgraduate Innovation Project(CXZZ13 0888)of Jiangsu Province
文摘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.
