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We solve explicitly the geodesic equation for a wide class of(pseudo)-Riemannian homogeneous manifolds (G/H,m), including those with Gcompact, as well as non-compact semisimple Lie groups, under a simple algebraiccondition for the metric m. We prove that these manifolds are geodesicallycomplete and their geodesics are orbits in G/H of a product of N one-parametersubgroups of G, where N is a natural number. This is intimately related to thefact that the metrics m can be regarded as N-parameter deformations of aRiemannian metric in G/H with special symmetries. We show that there exists awealth of metrics having the aforementioned type of geodesics; those metricsare constructed from any Lie subgroup series of the form H < H_1 < ... <H_{N-1} < G.

Geodesics as deformations of one-parameter subgroups in homogeneous manifolds