Lie Ball as Tangent Space to Poincare Space

Details Lie Ball as Tangent Space to Poincare Space Lie Ball as Tangent Space to Poincare Space

2006-12-06

arxiv.org/abs/math/0612155v1

Mathematics

Geometric Topology

Scientific paper

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 k\"ahler metric on the tangent space $TM$. The equivariant map has a simple geometric interpretation.

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