manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
A hyperbolic space is the analog of a Euclidean space as one passes from Euclidean geometry to hyperbolic geometry. The generalization of the concept of hyperbolic plane to higher dimension.
A hyperbolic manifold is a geodesically complete Riemannian manifold of constant sectional curvature .
Of particular interest are hyperbolic 3-manifolds.
Every hyperbolic manifold is a conformally flat manifold.
(e.g. Long-Reid 00, p. 4)
There are canonical zeta functions associated with suitable (odd-dimensional) hyperbolic manifolds, see at Selberg zeta function and Ruelle zeta function.
The Mostow rigidity theorem states that every hyperbolic manifold of dimension and of finite volume is uniquely determined by its fundamental group.
A Riemannian manifold
with zero sectional curvature is a Euclidean manifold?;
with +1 sectional curvature is an elliptic manifold?
See also
Textbook accounts:
John Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer 2006 (doi:10.1007/978-0-387-47322-2, pdf)
Michael Kapovich, Hyperbolic Manifolds and Discrete Groups, Modern Birkhäuser Classics, Birkhäuser 2008 (doi:10.1007/978-0-8176-4913-5)
See also
Darren D. Long, A. W. Reid, On the geometric boundaries of hyperbolic 4-manifolds (arXiv:math/0007197)
Wikipedia, Hyperbolic manifold
Wikipedia, Hyperbolic space
Last revised on July 21, 2020 at 17:51:36. See the history of this page for a list of all contributions to it.