Gaussian thermostats as geodesic flows of nonsymmetric linear connections
type: Item
instance of: scholarly article
title: Gaussian thermostats as geodesic flows of nonsymmetric linear connections
zbMATH Open document ID: 1168.53041
author: Piotr Przytycki
author: Maciej P. Wojtkowski
published in: Communications in Mathematical Physics
publication date: 2009-03-26T00:00:00Z
review text: Consider, on the unit tangent bundle \(T^1M\) of a compact Riemannian manifold \(M\), the equation \[ {du\over dt}= v;\quad {\nabla v\over dt}= |v|^2 E- \langle E,v\rangle v, \] where \(\nabla\) denotes the Levi-Cività connection, and \(E\) a given vector field on \(M\). This equation can be viewed as that of geodesics, with respect to other connections than \(\nabla\). The main result is that among these, \[ \widetilde\nabla_X Y:= \nabla_X Y-\langle X,Y\rangle E+\langle Y,E\rangle X \] defines the only one which is metric and has its torsion \(\widetilde T(X,Y)\) spanned (at each point) by \(\{X,Y\}\). It is also shown that, for any given metric connection, if the sectional curvature is \(< -{1\over 4}|T(.,.)|^2\), then the geodesic flow is Anosov.
reviewed by: Jacques Franchi
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 53B21
zbMATH DE Number: 5533167
zbMATH Keywords: geodesic flow
zbMATH Keywords: metric connections
zbMATH Keywords: Anosov flow
zbMATH Keywords: hyperbolicity
MaRDI profile type: Publication
full work available at URL: s00220-007-0376-2
OpenAlex ID: W1973997949
cites work: Q3277071
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DOI: 10.1007/S00220-007-0376-2
description: scientific article; zbMATH DE number 5533167