Analytic torsions on contact manifolds

Rumin, Michel; Seshadri, Neil

Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 727-782.

Nous définissons et étudions la torsion analytique du complexe de contact sur les variétés de contact. Nous montrons qu’elle coïncide avec la torsion de Ray–Singer sur les variétés CR de Seifert munies d’une représentation unitaire. Nous la calculons dans ces cas et l’exprimons à l’aide de propriétés dynamiques du flot de Reeb. En fait, notre fonction spectrale de torsion analytique coïncide avec une fonction zêta dynamique naturelle. Ces formules de trace «  à la Selberg  » persistent ici pour des métriques de courbure non constante sur la base.

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray–Singer torsion on any $3$-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-like trace formulae, that hold also in variable curvature.

Classification : 58J52, 32V05, 32V20, 11M36, 37C30
Keywords: analytic torsion, contact complex, CR Seifert manifold, trace formula
Mot clés : torsion analytique, complexe de contact, variété CR de Seifert, formule de trace

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