manifolds and cobordisms
cobordism theory, Introduction
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 $(X,g)$ of constant sectional curvature $-1$.
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 $\geq 3$ 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 account:
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 May 22, 2019 at 13:29:27. See the history of this page for a list of all contributions to it.