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

Chiakuei Peng; Zizhou Tang
January 2010
Acta Mathematica Sinica;Jan2010, Vol. 26 Issue 1, p25
Academic Journal
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( $$ \tilde e $$) = deg A, where e and $$ \tilde e $$ 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 M3 → E4, respectively. Consequently we find many metrics on S3 (Berger spheres) so that they can not be conformally embedded in E4.


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