nLab
hyperbolic manifold

Contents

Context

Riemannian geometry

Manifolds and cobordisms

Contents

Idea

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)(X,g) of constant sectional curvature 1-1.

Of particular interest are hyperbolic 3-manifolds.

Properties

Conformal flatness

Every hyperbolic manifold is a conformally flat manifold.

(e.g. Long-Reid 00, p. 4)

Zeta functions

There are canonical zeta functions associated with suitable (odd-dimensional) hyperbolic manifolds, see at Selberg zeta function and Ruelle zeta function.

Mostow rigidity theorem

The Mostow rigidity theorem states that every hyperbolic manifold of dimension 3\geq 3 and of finite volume is uniquely determined by its fundamental group.

Examples

A Riemannian manifold

  • with zero sectional curvature is a Euclidean manifold?;

  • with +1 sectional curvature is an elliptic manifold?

See also

References

Textbook account:

See also

Last revised on May 22, 2019 at 13:29:27. See the history of this page for a list of all contributions to it.