Convexly hyperbolic flows on unit tangent bundles of surfaces

We introduce a family of vector fields (convexly hyperbolic vector fields) on the unit tangent bundle of a compact surface of constant negative curvature, which generate flows that are semiconjugate to the geodesic flow. This $C^0$ open condition generalizes and simplifies the sufficient conditions obtained by Ratner [8]. We apply it to the class of tangential flows (“second order differential equations”).