Geodesic
Convexly hyperbolic flows on unit tangent bundles of surfaces
Informatics and Automation, Dynamical systems and related topics, Tome 216 (1997), pp. 373-383.

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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”). 
@article{TRSPY_1997_216_a25,
     author = {M. P. Wojtkowski},
     title = {Convexly hyperbolic flows on unit tangent bundles of surfaces},
     journal = {Informatics and Automation},
     pages = {373--383},
     publisher = {mathdoc},
     volume = {216},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_1997_216_a25/}
}
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M. P. Wojtkowski. Convexly hyperbolic flows on unit tangent bundles of surfaces. Informatics and Automation, Dynamical systems and related topics, Tome 216 (1997), pp. 373-383. http://geodesic.mathdoc.fr/item/TRSPY_1997_216_a25/