The Fried Conjecture for Morse-Smale Flows: A Survey on Ray-Singer and Milnor Metrics

Abstract

This paper reviews the generalised Fried conjecture for Morse-Smale flows on compact Riemannian manifolds. We first establish the proper framework by constructing the twisted de Rham complex and deriving the Hodge decomposition, which underpins the definition of the Ray-Singer torsion. On the dynamical side, we characterise Morse-Smale vector fields, introducing the Ruelle Zeta function to encode the spectral data of closed orbits and constructing the Thom-Smale complex to include the contribution of fixed points. These invariants are synthesised into the definition of the Milnor metric on the determinant line of the twisted cohomology. Finally, we present the theorem proving the conjecture: the Ray-Singer metric coincides with the Milnor metric, identifying the analytic torsion with the product of the Thom-Smale combinatorial torsion and the value at zero of the Ruelle Zeta function.