Geodesic
The existence of convex spherical metrics, all closed nonselfintersecting geodesics of which are hyperbolic
Izvestiya. Mathematics , Tome 14 (1980) no. 1, pp. 1-16

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In this paper it is shown that in any $C^1$-neighborhood of the standard metric $H_0$ on $S^2$, there exists a subset consisting of convex metrics, which is open in the $C^2$-topology, and all of whose closed nonselfintersecting geodesics are hyperbolic. Bibliography: 13 titles. 
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     title = {The existence of convex spherical metrics, all closed nonselfintersecting geodesics of which are hyperbolic},
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A. I. Gryuntal'. The existence of convex spherical metrics, all closed nonselfintersecting geodesics of which are hyperbolic. Izvestiya. Mathematics , Tome 14 (1980) no. 1, pp. 1-16. http://geodesic.mathdoc.fr/item/IM2_1980_14_1_a0/