Abstract
We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point of the Ruelle dynamical zeta function, even in the case where this value is not well defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang's Theorem.
Highlights
The study of the relation between the combinatorial/analytic torsion of a flat vector bundle and the Morse-Smale flow was initiated by Fried [Fri87] and Sánchez-Morgado [SM96]
– a spectral invariant: the Ray-Singer metric associated with a flat vector bundle with a Hermitian metric on a closed Riemannian manifold;
– a dynamical invariant: the Milnor metric which reflects the interactions between the fixed points and the closed orbits of the Morse-Smale flow, and generalizes the absolute value at zero point of the Ruelle dynamical zeta function;
Summary
The study of the relation between the combinatorial/analytic torsion of a flat vector bundle and the Morse-Smale flow was initiated by Fried [Fri87] and Sánchez-Morgado [SM96]. When compared with Bismut-Zhang’s theorem [BZ92, Th. 0.2], it seems natural to ask whether there is a relation between the torsion invariant (or more generally the Ray-Singer metric for non acyclic and non unitarily flat vector bundle) and a general Morse-Smale flow which has both fixed points and closed orbits. This is one of the motivations of Sánchez-Morgado’s work [SM96]. S.S. would like to thank Nguyen Viet Dang and Gabriel Rivière for fruitful discussions on Morse-Smale flows
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