Foundations of hyperbolic manifolds. (English) Zbl 0809.51001

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.


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)