The unit tangent bundle is a fundamental concept in differential geometry and is used in the study of manifolds, particularly in the context of differential geometry and geodesic flows. Given a smooth manifold \( M \), the unit tangent bundle, denoted as \( U(TM) \), consists of all unit tangent vectors at every point in \( M \).
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The unit tangent bundle of a Riemannian manifold is the sphere bundle of its tangent bundle. Its points are pairs with , , and .