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JOURNALS // Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica // Archive

Bul. Acad. Ştiinţe Repub. Mold. Mat., 2015 Number 2, Pages 3–20 (Mi basm389)

This article is cited in 1 paper

Survey articles

The Cotton tensor and Chern–Simons invariants in dimension $3$: an introduction

Sergiu Moroianu

Institutul de Matematică al Academiei Române, P.O. Box 1-764, RO-014700 Bucharest, Romania

Abstract: 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.

Keywords and phrases: Chern–Simons invariant, Schouten tensor, Cotton tensor, locally conformally flat metrics, eta invariant, Selberg zeta function of odd type.

MSC: 58J28, 53A30

Received: 19.09.2015

Language: English



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