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

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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(\( \tilde e \)) = degA, 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 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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Authors and Affiliations

  1. School of Mathematical Sciences, Graduate University, Chinese Academy of Sciences, Beijing, 100049, P. R. China

    Chiakuei Peng

  2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, P. R. China

    Zizhou Tang

Authors
  1. Chiakuei Peng
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  2. Zizhou Tang
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Correspondence to Zizhou Tang.

Additional information

Supported by NSFC (Grant Nos. 10531090 and 10229101) and Chang Jiang Scholars Program

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Peng, C., Tang, Z. Chern-Simons invariant and conformal embedding of a 3-manifold. Acta. Math. Sin.-English Ser. 26, 25–28 (2010). https://doi.org/10.1007/s10114-010-8559-8

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  • DOI: https://doi.org/10.1007/s10114-010-8559-8

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