= Unit tangent transitivity characterizes two-point homogeneity
For a connected <Riemannian manifold>, two-point homogeneity is equivalent to the <isometry> group being transitive on the <unit tangent bundle>. One direction follows by taking short equal-length radial <geodesic> segments: the <Gauss lemma> identifies their distance, and injectivity of the <exponential map> identifies the initial directions after the endpoints are matched. Conversely, unit tangent transitivity implies point homogeneity, hence completeness. The <Hopf-Rinow theorem> supplies minimizing <geodesics> for arbitrary equal-distance pairs. Matching their initial unit tangent vectors and using uniqueness of the <geodesic equation> matches their other endpoints.
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