Equivariant analytic torsions in de Rham theory
type: Item
instance of: scholarly article
zbMATH Open document ID: 0982.58022
DOI: 10.1016/S0764-4442(00)01772-9
publication date: 2002-03-25T00:00:00Z
Mathematics Subject Classification ID: 58J52
zbMATH DE Number: 1587963
zbMATH Keywords: equivariant analytic torsion
zbMATH Keywords: de Rham theory
zbMATH Keywords: \(V\)-invariant
author: Jean-Michel Bismut
author: Sebastian Goette
MaRDI profile type: Publication
full work available at URL: s0764-4442(00)01772-9
OpenAlex ID: W2079256467
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title: Equivariant analytic torsions in de Rham theory
published in: Comptes Rendus de l'Académie des Sciences. Série I. Mathématique
review text: Let \(F\) denote a Hermitian flat vector bundle on a compact Riemannian manifold \(X\). Let \(G\) denote a compact Lie group with an action on \(X\) by isometries that lifts to an action on \(F\) respecting both the metric and the connection. NEWLINENEWLINENEWLINEUnder these circumstances, the authors discuss and compare two definitions of equivariant analytic torsion in de Rham theory: the equivariant version of Ray-Singer torsion [\textit{D. B. Ray} and \textit{I. M. Singer}, Adv. Math. 7, 145-210 (1971; Zbl 0239.58014)] and their infinitesimal equivariant analytic torsion which is a renormalized version of the torsion suggested by \textit{J. Lott} [J. Funct. Anal. 125, No. 2, 181-236 (2000; Zbl 0824.58044)] and imitates the construction of Chern analytic torsion forms in de Rham theory [\textit{J.-M. Bismut} and \textit{J. Lott}, J. Am. Math. Soc. 8, No. 2, 291-363 (1995; Zbl 0837.58028)]. The difference between these two torsions is then expressed as the integral of local quantities. One of these is the apparently new \(V\)-invariant of odd-dimensional \(G\)-manifolds. Moreover, the difference formula sheds new light on formulas due to \textit{U. Bunke} [Geom. Funct. Anal. 9, No. 1, 67-89 (1999; Zbl 0929.57013), and Am. J. Math. 122, 377-401 (2000)] relating the torsion forms to the equivariant Euler characteristic of the manifold \(X\).
reviewed by: Martin Raussen
description: scientific article