Chern-Simons invariant and conformal embedding of a 3-manifold |
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| Authors: | Chiakuei Peng Zizhou Tang |
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| Institution: | 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
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| 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) — CS($
\tilde e
$
\tilde e
) = degA, where e and $
\tilde e
$
\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 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. |
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| Keywords: | Chern-Simons invariant Berger sphere conformal embedding |
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