TITLE

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

AUTHOR(S)
Chiakuei Peng; Zizhou Tang
PUB. DATE
January 2010
SOURCE
Acta Mathematica Sinica;Jan2010, Vol. 26 Issue 1, p25
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
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) â€” CS( $$\tilde e$$) = deg A, where e and $$\tilde e$$ are two (global) frames of M, and A: M â†’ SO(3) is the â€œdifferenceâ€ map. An interesting phenomenon is that the â€œjumpsâ€ 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( eâˆ’), where e+ and eâˆ’ are the â€œleftâ€ and â€œrightâ€ quaternionic frames on M3 induced from an immersion M3 â†’ E4, respectively. Consequently we find many metrics on S3 (Berger spheres) so that they can not be conformally embedded in E4.
ACCESSION #
47481531

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