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
Academic Journal
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) — 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.
ACCESSION #
47481531

 

Related Articles

  • Knots and classical 3-geometries. Toh, Tze-Chuen; Anderson, Malcolm R. // Journal of Mathematical Physics;Jan1995, Vol. 36 Issue 1, p596 

    Presents an exact statement of Rovelli's conjecture that there is a correspondence between the space of link classes of Riemannian 3-manifold and the space of 3-geometries (on the same manifold). Verification of the conjecture for the case when the 3-manifold is compact, orientable, and closed;...

  • Two Forms of Stability Change for Integral Manifolds. Shchepakina, E. A. // Differential Equations;May2004, Vol. 40 Issue 5, p766 

    Describes two forms of stability change for integral manifolds. Features of the use of attracting-repulsive integral manifolds analogus to duck trajectories for the modeling and control of critical combustion modes; Analysis of the existence of continuous attracting-repulsive integral surfaces...

  • Generic metrics and the mass endomorphism on spin three-manifolds. Hermann, Andreas // Annals of Global Analysis & Geometry;Feb2010, Vol. 37 Issue 2, p163 

    Let ( M, g) be a closed Riemannian spin manifold. The constant term in the expansion of the Green function for the Dirac operator at a fixed point $${p\in M}$$ is called the mass endomorphism in p associated to the metric g due to an analogy to the mass in the Yamabe problem. We show that the...

  • Generalized Einstein conditions on holomorphic Riemannian manifolds. Olszak, K.; Olszak, Z. // Acta Mathematica Hungarica;Dec2006, Vol. 113 Issue 4, p345 

    At first, a necessary and sufficient condition for a K�hler-Norden manifold to be holomorphic Einstein is found. Next, it is shown that the so-called (real) generalized Einstein conditions for K�hler-Norden manifolds are not essential since the scalarcurvature of such manifolds is...

  • First Eigenvalue, Volume Growth and Ricci Curvature. Etemad, Azam; Samea, Hojatolah // Southeast Asian Bulletin of Mathematics;2005, Vol. 29 Issue 4, p715 

    We first prove a new relation between the existence of an upper bound for volume growth in complete Riemannian manifold with the existence of a lower hound for its Ricci curvature. Then we prove that if the Ricci curvature outside the some compact subset of a compelet, non compact Riemannian...

  • Vrănceanu connections and foliations with bundle-like metrics. Bejancu, Aurel; Farran, Hani Reda // Proceedings of the Indian Academy of Sciences: Mathematical Scie;Feb2008, Vol. 118 Issue 1, p99 

    We show that the Vrănceanu connection which was initially introduced on non-holonomic manifolds can be used to study the geometry of foliated manifolds. We prove that a foliation is totally geodesic with bundle-like metric if and only if this connection is a metric one. We introduce the...

  • CURVATURE COMPUTATIONS OF 2-MANIFOLDS IN IR[supk]. Guo-liang Xu; Bajaj, Chandrajit L. // Journal of Computational Mathematics;Sep2003, Vol. 21 Issue 5, p681 

    In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2dimensional Riemannian manifold embedded in R[supk] with k = 3.

  • Averaging of Jacobi fields along geodesics on manifolds of random curvature. Grachev, D. // Journal of Mathematical Sciences;Jul2009, Vol. 160 Issue 1, p128 

    The paper considers the Jacobi field along a geodesic on a Riemannian manifold on which the curvature is a stochastic process. The author introduces the concept of linearizing tensor of the Jacobi field on the basis of which a sufficiently universal averaging algorithm is constructed. The...

  • A differentiable sphere theorem on manifolds with reverse volume pinching. Wang, P. // Acta Mathematica Hungarica;Apr2008, Vol. 119 Issue 1/2, p63 

    Based on the celebrated 1/4-pinching sphere theorem, we prove a differentiable sphere theorem on Riemannian manifolds with reverse volume pinching.

Share

Read the Article

Courtesy of THE LIBRARY OF VIRGINIA

Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics