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

作　　者：Chiakuei PENG Zizhou TANG 

机构地区：[1]School of Mathematical Sciences, Graduate University, Chinese Academy of Sciences, Beijing 100049, P. R. China [2]School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2010年第1期25-28,共4页数学学报（英文版）

基　　金：Supported by NSFC (Grant Nos. 10531090 and 10229101);Chang Jiang Scholars Program

摘　　要：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.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 

分 类 号：O186.12[理学—数学] O1-09[理学—基础数学]