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Chern-Simons Invariant and Conformal Embedding of a 3-Manifold

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Acta Mathematica Sinica,English Series 2010年第1期26卷 25-28页

作者： Chiakuei PENG zizhou tang School of Mathematical Sciences Graduate University Chinese Academy of Sciences Beijing 100049 P. R. China School of Mathematical Sciences Laboratory of Mathematics and Complex Systems Beijing Normal University Beijing 100875 P. R. China

This note studies the Chern-Simons invariant of a closed oriented Riemannian 3-manifold M. The first achievement is to establish the formula CS（e） - CS（e） = degA, where e and e are two （global） frames of M, and A ： M → SO（3） is the ＂difference＂ map. An interesting phenomenon is that the ＂jumps＂ of the Chern-Simons integrals for various frames of many 3-manifolds are at least two, instead of one. The second purpose is to give an explicit representation of CS（e＋） and CS（e_）, where e＋ and e_ are the ＂left＂ and ＂right＂ quaternionic frames on M3 induced from an immersion M^3 → E^4, respectively. Consequently we find many metrics on S^3 （Berger spheres） so that they can not be conformally embedded in E^4.

关键词： Chern-Simons invariant Berger sphere conformal embedding

Topology and Curvature of Isoparametric Families in Spheres

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Communications in Mathematics and Statistics 2023年第2期11卷 439-475页

作者： Chao Qian zizhou tang Wenjiao Yan School of Mathematics and Statistics Beijing Institute of TechnologyBeijing 100081People’s Republic of China Chern Institute of Mathematics and LPMC Nankai UniversityTianjin 300071People’s Republic of China School of Mathematical Sciences Laboratory of Mathematics and Complex SystemsBeijing Normal UniversityBeijing 100875People’s Republic of China

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal *** present paper has two *** first part investigates topology of the isoparametric families,namely the homotopy,homeomorphism,or diffeomorphism types,parallelizability,as well as the Lusternik-Schnirelmann *** part extends substantially the results of Wang(J Differ Geom 27:55-66,1988).The second part is concerned with their curvatures;more precisely,we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

关键词： Isoparametric hypersurface Focal submanifold Homotopy equivalent Homeomorphism Diffeomorphism Parallelizability Lusternik-Schnirelmann category Sectional curvature Ricci curvature

Integrability Condition on an Almost Complex Structure and Its Application

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Acta Mathematica Sinica,English Series 2005年第6期21卷 1459-1464页

作者： Chia Kuei PENG Zi Zhou tang(zizhou tang) Department of Mathematics Graduate School The Chinese Academy of Sciences Beijing 100039 P. R. China School of Mathematical Sciences Beijing Normal University Beifing 100875 P. R. China Department of Mathematical Sciences Tsinghua University Beijing 100084 P. R. China

We obtain and apply a simple form of the integrability condition of an almost complex structure to the Hopf manifolds.

关键词： almost complex structure integrability

ISOPARAMETRIC HYPERSURFACES AND COMPLEX STRUCTURES

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Acta Mathematica Scientia 2022年第6期42卷 2223-2229页

作者： zizhou tang Wenjiao YAN Chern Institute of Mathematics&LPMC Nankai UniversityTianjin 300071China School of Mathematical Sciences Laboratory of Mathematics and Complea SystemsBeijing Normal UniversityBeijing 100875China

The main purpose of this note is to construct almost complex or complex structures on certain isoparametric hypersurfaces in unit *** a consequence,complex structures on S^(1)×S^(7)×S^(6),and on S^(10)×S^(3)×S(2)w... 详细信息

关键词： almost complex structure integrable Hermitian structure isoparametric hypersurface