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THE UNIT BUNDLE OF A REAL HYPERBOLIC SPACE
Keywords:
hyperbolic space, isotropy representations, unit bundleDOI:
https://doi.org/10.17654/0972415X24007Abstract
The purpose of this article is to study the two homogeneous structures of the unit bundle $U H^n$ of a real hyperbolic space, namely
$$
U H^n \simeq S O_0(1, n) / S O(n-1)
$$
and
$$
U H^n \simeq S O_0(1, n) \times S O_0(1,1) / S O(n-1) \times S O_0(1,1)
$$
In both the cases, we determine the $G$-invariant Riemannian metrics. In passing, we examine whether the geodesic flow is an isometry of $U H^n$ when equipped with its Levi-Civita metric.
Finally, we study the manifold of geodesics of $H^n$ seen as a homogeneous space.
Received: April 11, 2024
Revised: June 5, 2024
Accepted: September 12, 2024
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