Chern-Simons invariant and conformal embedding of a 3-manifold

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 M 3 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.