# 6 Lie Ball as Tangent Space to Poincaré Ball

• Tchangang Tambekou
• Published 2006

#### Abstract

We equip the whole tangent space TM to a hyperbolic manifold M (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of M extend to isometries of TM by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In the case of hyperbolic space, we exhibit an equivariant diffeomorphism between TM and the fourth symmetric complex domain of E. Cartan, also known as the Lie ball. The closure of the Lie ball appears as a horospheric compactification of the tangent bundle to hyperbolic space, and its Bergmann metric gives an intrinsic natural kähler metric on the tangent space TM . The equivariant map has a simple geometric interpretation. We propose hereafter another complexification of the hyperbolic space, at least as ’natural’ as the one given by the Akhiezer-Gindikin domain. The result is still the Lie ball, but here, not only the complex structure is global, but the leaves of the riemannian foliation are complete hyperbolic disks instead of having partial flat structure. This leads to an equivariant compactification of hyperbolic space. We define a diffeomorphism between the Lie ball and the tangent space to the Poincaré ball, and pull back the Kähler symmetric structure of the Lie ball to the tangent space to hyperbolic space. In the sequel, n denotes an integer greater or equal to 2. Main result. The action of the hyperbolic group extends to the closure Bn of the Lie ball as isometries, by holomorphic continuation. There exists a diffeomorphism θ between TB, the tangent space to the hyperbolic space, and the Lie ball B, with the following properties: 1. θ is equivariant for the action of the hyperbolic group. 2. Bn is an equivariant compactification of the tangent space TB by codimension 2 horospheres. 3. The tangent space to every geodesic of the hyperbolic space is totally geodesic. Its image by θ is isometric to a hyperbolic disk. 4. Any vector line of the tangent space at a point is a geodesic. These past twenty years, serious efforts have been undertaken to equip the tangent space to a hyperbolic manifold with a kähler structure. Several extensions to TM of the metric on M have been studied. The most famous

### Cite this paper

@inproceedings{Tambekou20066LB, title={6 Lie Ball as Tangent Space to Poincar{\'e} Ball}, author={Tchangang Tambekou}, year={2006} }