\(W\)-flows on Weyl manifolds and Gaussian thermostats
This is a very clearly written paper on Weyl structures, including examples. A Weyl structure on a compact Riemannian \(n\)-dimensional manifold \(M\) consists of a family of torsion-free connections with the parallel transport preserving conformal class of metrics. The author fixes the Riemannian metric, which plays the role of a physical space, and takes a Weyl flow on \(UM\), whose orbits are geodesics of the Weyl connection. The Weyl structure coincides with the Levi-Civita connection on a rescaled metric. He calls \(W\)-flows those defined by this new metric. He points out several physical settings where this situation holds: on the cotangent bundle, in the Hamiltonian formulation of Gaussian thermostats, which are models in nonequilibrium statistical mechanics. Also, in conformally symplectic formulations of isokinetic dynamics. The author then uses the Jacobi equation for the geodesic flow of a Weyl connection and straightforwardly obtains the linear approximation of the flow in terms of the curvature of the Weyl connection. The symmetric component is a multiple of the identity but the antisymmetric part corresponds to sectional curvatures in plane directions. By analysing their sign he obtains that sectional curvatures for the Weyl structures defined by a constant vector field on a flat two dimensional torus are zero, but this is not the case in higher dimensions. He then turns to study extensively the hyperbolic properties of \(W\)-flows. Finally, taking a Riemannian manifold \(M\) with boundary, he analyses a \(W\)-flow in two dimensions with collisions, as defined in [the author and \textit{P. Maciej}, Commun. Math. Phys. 126, 507-533 (1990; Zbl 0695.70007)], and observes that the splitting obtained for this case, does not occur in dimensions 3 and higher. For other articles on Weyl manifolds, see \textit{A. Yoshioka} [Rep. Math. Phys. 43, No. 1-2, 357-366 (1999; Zbl 0957.53049) and Lobachevskii J. Math. 4, 177-206 (1999; Zbl 0963.53055)] for those on nonequilibrium statistical mechanics, see \textit{E. G. D. Cohen} and \textit{G. Gallavoti} [J. Stat. Phys. 96, No. 5-6, 1343-1349 (1999; Zbl 0938.82032)], \textit{D. Ruelle} [J. Stat. Phys. 95, No. 1-2, 393-468 (1999; Zbl 0934.37010)], \textit{C. P. Dettmann} [Encycl. Math. Sci. 101(2), 315-365 (2000; Zbl 1026.82018)], \textit{C. P. Dettmann} and \textit{G. P. Morris} [Hamiltonian formulations of the Gaussian isokinetic thermostat, Phys. Rev. E 54, 2495-2500 (1996)]. Finally, for papers on Lorentz gas, see \textit{D. Szász} [Encyclopaedia of Mathematical Sciences. Mathematical Physics 101(2). Berlin: Springer. viii, 458 p. (2000; Zbl 0953.00014)] and \textit{T. Tél} and \textit{J. Vollmer} [Encycl. Math. Sci. 101(2), 367-418 (2000; Zbl 0967.37018)].
scientific article; zbMATH DE number 1557019