A geodesic can be represented by a Hamiltonian vector field, and its flow is called a geodesic flow. Geodesics on Riemannian manifolds have been studied, and in particular Bolotin and Rabinowitz proved that there are geometrically distinct minimizing closed geodesics and homoclinic/heteroclinic orbits under some conditions.
In this talk, we consider geodesics on a perturbed surface of revolution. As well known, geodesic flows on surfaces of revolution are integrable and have periodic solutions which correspond to closed geodesics. We apply Melnikov's method to the perturbed system of the Hamiltonian vector field, and detect the transversality of the homoclinic/heteroclinic orbits.