Foundations of hyperbolic manifolds. (English) Zbl 0809.51001

Ratcliffe, John G.

Graduate Texts in Mathematics. 149. Berlin: Springer-Verlag. xi, 747 p. (1994).

A detailed and extensive study of geometric manifolds, esp. of hyperbolic ones, is preceded by an expose of foundations of non-Euclidean spaces, of their models and of related groups of transformations. The surfaces, 2- manifolds and 3-manifolds are treated. The expose culminates in a chapter on geometric orbifolds and Poincaré fundamental polyhedron theorem. Each of the 13 chapters is completed by a section ‘Historical notes’; the bibliography contains 422 references.

Reviewer: A.Szybiak (Kitchener)

MSC:

51-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry
57M50	General geometric structures on low-dimensional manifolds
51M20	Polyhedra and polytopes; regular figures, division of spaces
30F40	Kleinian groups (aspects of compact Riemann surfaces and uniformization)
32Q45	Hyperbolic and Kobayashi hyperbolic manifolds
51F15	Reflection groups, reflection geometries
51M10	Hyperbolic and elliptic geometries (general) and generalizations
52A20	Convex sets in \(n\) dimensions (including convex hypersurfaces)

Keywords:

Moebius transformations; polyhedra; space forms; gluing; reflection groups; discrete groups; Kleinian groups; Fuchsian; orbifolds; Poincaré fundamental polyhedron