Homogeneous and flow-invariant geometry on the unit tangent bundle of hyperbolic space

2026/07/18 by Daniel Koama, Léonard Todjihoundé
Mathematics · #math.DG

paper · pdf

Abstract

We construct the Sasaki metric on the unit tangent bundle U_g M of a Riemannian manifold (M , g) and describe the unit tangent bundle Uℍⁿ of real hyperbolic space as a homogeneous space, both under SO₀ (1, n) and under the larger group SO₀ (1, n) × SO₀ (1, 1), yielding explicit of G-invariant metrics. Using Hopf coordinates and Busemann functions, we then construct a Riemannian metric g_Hopf on Uℍⁿ that is invariant under the geodesic flow, and we identify the horospherical cylinders as totally geodesic leaves of a natural foliation associated to a Busemann function, with respect to an explicit metric connection with torsion.

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