Hyperbolic manifold

In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.

A perspective projection of a dodecahedral tessellation in H³. This is an example of what an observer might see inside a hyperbolic 3-manifold.

The Pseudosphere. Each half of this shape is a hyperbolic 2-manifold (i.e. surface) with boundary.

Rigorous Definition

A hyperbolic $n$ -manifold is a complete Riemannian $n$ -manifold of constant sectional curvature $-1$ .

Every complete, connected, simply-connected manifold of constant negative curvature $-1$ is isometric to the real hyperbolic space ${\mathbb {H}}^{n}$ . As a result, the universal cover of any closed manifold $M$ of constant negative curvature $-1$ is ${\mathbb {H}}^{n}$ . Thus, every such $M$ can be written as ${\mathbb {H}}^{n}/\Gamma$ where $\Gamma$ is a torsion-free discrete group of isometries on ${\mathbb {H}}^{n}$ . That is, $\Gamma$ is a discrete subgroup of $SO_{{1,n}}^{+}{\mathbb {R}}$ . The manifold has finite volume if and only if $\Gamma$ is a lattice.

Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and ends which are the product of a Euclidean ( $n-1$ )-manifold and the closed half-ray. The manifold is of finite volume if and only if its thick part is compact.

For $n>2$ the hyperbolic structure on a finite volume hyperbolic $n$ -manifold is unique by Mostow rigidity and so geometric invariants are in fact topological invariants.

References

Kapovich, Michael (2009) [2001], Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4913-5, ISBN 978-0-8176-4912-8, MR 1792613
Maclachlan, Colin; Reid, Alan W. (2003), The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, 219, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98386-8, MR 1937957
Ratcliffe, John G. (2006) [1994], Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, 149 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-47322-2, ISBN 978-0-387-33197-3, MR 2249478
Hyperbolic Voronoi diagrams made easy, Frank Nielsen

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Hyperbolic manifold

Rigorous Definition

See also

References