The geodesic flow on a homogeneous space with an invariant metric can be naturally considered within the framework of Smale's mechanical systems with symmetry. In this way we have at our disposal the whole method of Smale for studying such systems. We prove that certain sets 2', 2, Im, o and Re which play an important role in the global behavior of those systems, have a particularly simple structure in our case, and we also find some geometrical implications about the geodesies. The results

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... ined are especially powerful for the case of Lie groups, as in the rigid body problem.