## Complex hyperbolic geometry.(English)Zbl 0939.32024

Oxford Mathematical Monographs. Oxford Science Publications. Oxford: Clarendon Press. xx, 316 p. (1999).
This book treats the geometry of the complex hyperbolic space $$H^n_c$$ and its boundary $$\partial H^n_c$$ in a fairly comprehensive manner. Several models of $$H^n_c$$ are considered, the projective model being the primary one. The chapters are headlined as follows.
1. The complex projective line
2. Algebraic and geometric background
3. The Ball model
4. The paraboloid model and Heisenberg geometry
5. Bisectors and spinal spheres
6. Automorphisms
7. Numerical invariants
8. Extors in Projective Space
9. Intersections of bisectors.
Some emphasis is laid on geometrical invariants and on the trigonometry of the $$H^n_c$$. The author boldly introduces new short names into complex geometry: bisector (formerly called equidistant hypersurface or spinal hypersurface), spinal sphere (the boundary in $$\partial H^n_c$$ of a bisector), extor (projective extension of a bisector), and more. There are relations to Riemann, Kähler, Heisenberg, symplectic, and contact geometrics.
Explanations try to make the book self-consistent. Some richness of details and many exercises give the text a didactical character. Computer graphics visualise some of the abstractly defined surfaces. The bibliographical list has 175 entries. An appendix shortly discusses Georges Giraud’s papers of 1971-1921 on automorphic functions.

### MSC:

 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 53C40 Global submanifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds

### Keywords:

complex space; hyperbolic space; projective space; Kähler space