Gaussian thermostats as geodesic flows of nonsymmetric linear connections (Q1006850)

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.