Geodesic
The Cotton tensor and Chern–Simons invariants in dimension $3$: an introduction
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2015), pp. 3-20 
Sergiu Moroianu. The Cotton tensor and Chern–Simons invariants in dimension $3$: an introduction. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2015), pp. 3-20. http://geodesic.mathdoc.fr/item/BASM_2015_2_a0/
@article{BASM_2015_2_a0,
     author = {Sergiu Moroianu},
     title = {The {Cotton} tensor and {Chern{\textendash}Simons} invariants in dimension~$3$: an introduction},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {3--20},
     year = {2015},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2015_2_a0/}
}
TY  - JOUR
AU  - Sergiu Moroianu
TI  - The Cotton tensor and Chern–Simons invariants in dimension $3$: an introduction
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2015
SP  - 3
EP  - 20
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/BASM_2015_2_a0/
LA  - en
ID  - BASM_2015_2_a0
ER  - 
%0 Journal Article
%A Sergiu Moroianu
%T The Cotton tensor and Chern–Simons invariants in dimension $3$: an introduction
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2015
%P 3-20
%N 2
%U http://geodesic.mathdoc.fr/item/BASM_2015_2_a0/
%G en
%F BASM_2015_2_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

We review, with complete proofs, the theory of Chern–Simons invariants for oriented Riemannian $3$-manifolds. The Cotton tensor is the first-order variation of the Chern–Simons invariant. We deduce that it is conformally invariant, and trace- and divergence-free, from the corresponding properties of the Chern–Simons invariant. Moreover, the Cotton tensor vanishes if and only if the metric is locally conformally flat. In the last part of the paper we survey the link of Chern–Simons invariants with the eta invariant and with the central value of the Selberg zeta function of odd type.

[1] M. F. Atiyah, V. K. Patodi, I. M. Singer, “Spectral asymmetry and Riemannian geometry. I”, Math. Proc. Cambridge Philos. Soc., 77 (1975), 4–69 | DOI | MR

[2] M. F. Atiyah, V. K. Patodi, I. M. Singer, “Spectral asymmetry and Riemannian geometry. II”, Math. Proc. Cambridge Philos. Soc., 78 (1975), 405–432 | DOI | MR | Zbl

[3] A. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, Berlin, 1987 | MR | Zbl

[4] S.-S. Chern, J. Simons, “Characteristic forms and geometric invariants”, Ann. Math. (2), 99 (1974), 48–69 | DOI | MR | Zbl

[5] É. Cotton, “Sur les variétés à trois dimensions”, Toulouse Ann. (2), 1 (1899), 385–438 | DOI | MR | Zbl

[6] C. Guillarmou, S. Moroianu, “Chern-Simons line bundle on Teichmüller space”, Geometry Topology, 18 (2014), 327–377 | DOI | MR | Zbl

[7] J. Millson, “Closed geodesics and the $\eta$-invariant”, Ann. Math. (2), 108:1 (1978), 1–39 | DOI | MR | Zbl

[8] A. Selberg, “Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series”, J. Indian Math. Soc. (N.S.), 20 (1956), 47–87 | MR | Zbl

[9] E. Stiefel, “Richtungsfelder und Fernparallelismus in $n$-dimensionalen Mannigfaltigkeiten”, Comment. Math. Helv., 8:1 (1935), 305–353 | DOI | MR

[10] R. Thom, “Espaces fibrés en sphères et carrés de Steenrod”, Ann. Sci. Ecole Norm. Sup., 69 (1952), 109–182 | MR | Zbl

[11] E. Witten, “Quantum field theory and the Jones polynomial”, Commun. Math. Phys., 121 (1989), 351–399 | DOI | MR | Zbl