AI 中文总结
研究双曲空间单位切丛上的几何，通过构造萨斯卡度量等，将其描述为齐次空间并得到\(G\)-不变度量，利用霍普夫坐标和布塞曼函数构造测地线流不变的黎曼度量，识别水平圆柱为全测地叶，给出显式度量联络。
AI 中文摘要
我们在黎曼流形\((M, g)\)的单位切丛\(U_g M\)上构造了萨斯卡度量，并将实双曲空间的单位切丛\(U\mathbb{H}^n\)描述为齐次空间，在\(SO_0 (1, n)\)以及更大的群\(SO_0 (1, n) \times SO_0 (1, 1)\)作用下，得到了\(G\)-不变度量的显式形式。利用霍普夫坐标和布塞曼函数，我们在\(U\mathbb{H}^n\)上构造了一个关于测地线流不变的黎曼度量\(g_{Hopf}\)，并将水平圆柱识别为与布塞曼函数相关的自然叶状结构的全测地叶，这是相对于一个具有挠率的显式度量联络而言的。
英文摘要
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 $U\mathbb{H}^n$ of real hyperbolic space as a homogeneous space, both under $SO_0 (1, n)$ and under the larger group $SO_0 (1, n) \times SO_0 (1, 1)$, yielding explicit of $G$-invariant metrics. Using Hopf coordinates and Busemann functions, we then construct a Riemannian metric $g_{Hopf}$ on $U\mathbb{H}^n$ 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.
