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. PREPARATION AND LEMMASWe start recalling theMain Theorem of Abresch. Given an isoparametric hypersurface in S<sup>n+1</sup> with g= 4, then the pair （m<sub>,</sub> m<sub>+</sub>...I. PREPARATION AND LEMMASWe start recalling theMain Theorem of Abresch. Given an isoparametric hypersurface in S<sup>n+1</sup> with g= 4, then the pair （m<sub>,</sub> m<sub>+</sub> ）-w. r. g, we may assume that m<sub> </sub>≤m<sub>+</sub>, satisfies one of the three conditions below:展开更多
I. INTRODUCTIONLet S<sup>2n+1</sup> be the （2n+1）- dimensional standard sphere in complex （n+1） space C<sup>n+1</sup>. Let T: S<sup>2+1</sup>→S<sup>2n+1</sup> be th...I. INTRODUCTIONLet S<sup>2n+1</sup> be the （2n+1）- dimensional standard sphere in complex （n+1） space C<sup>n+1</sup>. Let T: S<sup>2+1</sup>→S<sup>2n+1</sup> be the transformation defined by T（z<sub>0</sub>, z<sub>1</sub>, …, z<sub>n</sub>） = （e （2πi）/p Z<sub>0</sub>, e （2πi）/p Z<sub>1</sub>, …, e （2πi）/p z<sub>n</sub>）, where Z<sub>0</sub>, Z<sub>1</sub>, …, Z<sub>n</sub> are complex numbers with. T acts freely on S<sup>2n+1</sup> and generates a cyclic group Z<sub>p</sub> of order p, and the orbit space is a standard Lens space L<sup>n</sup>（p）.展开更多
1 Introduction Let K be a CW-complex and L be its subcomplex. A real (resp. complex) vector bundle over L is said to be extendible to K if it is equivalent to the restriction of a real (resp. complex) vector bundle ov...1 Introduction Let K be a CW-complex and L be its subcomplex. A real (resp. complex) vector bundle over L is said to be extendible to K if it is equivalent to the restriction of a real (resp. complex) vector bundle over K. R. L. E. Schwarzenberger studied the non-extendibility of vector bundles over CP^n (resp. RP^n) to CP^m(resp. RP^m), m】n, where CP^n (resp. RP^n) is the complex (resp. real)展开更多
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
Let S<sup>α</sup> be the α-dimensional standard sphere in （α+1）-dimensional Euclidean space R<sup>α+1</sup>. The classical Borsuk-Ulam theorem asserts that if there exists a continuous ma...Let S<sup>α</sup> be the α-dimensional standard sphere in （α+1）-dimensional Euclidean space R<sup>α+1</sup>. The classical Borsuk-Ulam theorem asserts that if there exists a continuous map f: S<sup>m</sup>→S<sup>n</sup>, such that f（-x） = -f（x） is satisfied for all x∈S<sup>m</sup>, then m≤n.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.
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.
基金partially supported by NSFC(No.12171037)the Fundamental Research Funds for the Central Universities+5 种基金partially supported by NSFC(Nos.12171037,12271040)China Postdoctoral Science Foundation(No.2022M720261)partially supported by NSFC(Nos.11931007,11871282)Nankai Zhide Foundation and Tianjin Outstanding Talents FoundationChina Postdoctoral Science Foundation(No.BX20230018)National Key R&D Program of China(No.2020YFA0712800)。
文摘I. PREPARATION AND LEMMASWe start recalling theMain Theorem of Abresch. Given an isoparametric hypersurface in S<sup>n+1</sup> with g= 4, then the pair （m<sub>,</sub> m<sub>+</sub> ）-w. r. g, we may assume that m<sub> </sub>≤m<sub>+</sub>, satisfies one of the three conditions below:
文摘I. INTRODUCTIONLet S<sup>2n+1</sup> be the （2n+1）- dimensional standard sphere in complex （n+1） space C<sup>n+1</sup>. Let T: S<sup>2+1</sup>→S<sup>2n+1</sup> be the transformation defined by T（z<sub>0</sub>, z<sub>1</sub>, …, z<sub>n</sub>） = （e （2πi）/p Z<sub>0</sub>, e （2πi）/p Z<sub>1</sub>, …, e （2πi）/p z<sub>n</sub>）, where Z<sub>0</sub>, Z<sub>1</sub>, …, Z<sub>n</sub> are complex numbers with. T acts freely on S<sup>2n+1</sup> and generates a cyclic group Z<sub>p</sub> of order p, and the orbit space is a standard Lens space L<sup>n</sup>（p）.
文摘1 Introduction Let K be a CW-complex and L be its subcomplex. A real (resp. complex) vector bundle over L is said to be extendible to K if it is equivalent to the restriction of a real (resp. complex) vector bundle over K. R. L. E. Schwarzenberger studied the non-extendibility of vector bundles over CP^n (resp. RP^n) to CP^m(resp. RP^m), m】n, where CP^n (resp. RP^n) is the complex (resp. real)
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.
文摘Let S<sup>α</sup> be the α-dimensional standard sphere in （α+1）-dimensional Euclidean space R<sup>α+1</sup>. The classical Borsuk-Ulam theorem asserts that if there exists a continuous map f: S<sup>m</sup>→S<sup>n</sup>, such that f（-x） = -f（x） is satisfied for all x∈S<sup>m</sup>, then m≤n.
基金Project supported partially by the National Natural Science Foundation of China (Grant No. 19531050)the State Education Commission Foundation of China.
文摘Two non-existence theorems on harmonic polynomial morphisms between Euclidean spaces have been
基金This work was supported partially by the Hong Kong Qiu-Shi Foundation, the Outstanding Youth Foundation of China and the Education Foundation of Tsinghua University, as well as the Grants-in-Aid for Science Research of Japanese Ministry of Education.
文摘By using the bordism group, this paper provides an alternative proof of Weiping Zhang's theorem on counting Kervaire semi-characteristic.
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.
