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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold M is the projection of a closed orbit of the geodesic flow on M.On the unit sphere, every great circle is an example of a closed geodesic. On a compact hyperbolic surface, closed geodesics are in one-to-one correspondence with non-trivial conjugacy classes of elements in the Fuchsian group of the surface. A prime geodesic is an example of a closed geodesic.It defines a Hamiltonian flow on (co)tangent bundle with the (pseudo-)Riemannian metric as the Hamiltonian. In particular it preserves the (pseudo-)Riemannian metric g, i.e.