Let be a (complete) Riemannian surface, and let be an open subset whose closure is homeomorphic to a disk. We prove that if is smooth and it satisfies a strong concavity assumption, then there are at least two distinct orthogonal geodesics in . Using the results given in Giamb et al. (Adv Differ Eq 10:931-960, 2005), we then obtain a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. In our proof we shall use recent deformation results proved in Giamb et al. (Nonlinear Anal Ser A Theory Methods Appl 73:290-337, 2010).
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机译：令为（完整的）黎曼曲面，并令其为封闭的圆盘同胚的开放子集。我们证明，如果光滑且满足强凹度假设，则中至少存在两个不同的正交测地线。使用Giamb等人给出的结果。 （Adv Differ Eq 10：931-960，2005），我们然后得到一类哈密顿系统存在两个不同的制动轨道的证明。在我们的证明中，我们将使用Giamb等人证明的最新变形结果。 （非线性肛门Ser A Theory Methods Appl 73：290-337，2010）。
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