Hyperbolic manifold

In mathematics , hyperbolic manifolds are Riemannian manifolds with constant negative curvature . They play an important role in the low-dimensional topology, especially in Thurston's geometry program.

definition

A hyperbolic manifold is a complete Riemannian manifold with constant intersection curvature . (A Riemannian metric with constant sectional curvature is called a hyperbolic metric . A hyperbolic manifold is therefore a manifold with a complete hyperbolic metric.) ${\displaystyle M}$ ${\displaystyle -1}$ ${\displaystyle -1}$

Equivalent definition 1: A hyperbolic manifold is a Riemannian manifold whose universal superposition is isometric to the hyperbolic space .

Equivalents Definition 2: A hyperbolic manifold is a Riemannian manifold of the mold , wherein the hyperbolic space and a discrete subgroup is the group of isometric drawings of hyperbolic space. ${\displaystyle M}$ ${\displaystyle \Gamma \backslash \mathbb {H} ^{n}}$ ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle \Gamma \subset \operatorname {Isom} (\mathbb {H} ^{n})}$

Hyperbolic monodromia

Because the hyperbolic space contractible is, must the definition used in two group isomorphic to the fundamental group to be. The resulting from Definition 2 representation is also called Monodromiedarstellung or hyperbolic monodromy referred. ${\displaystyle \Gamma }$ ${\displaystyle \pi _{1}M}$ ${\displaystyle \rho \colon \pi _{1}M\to \operatorname {Isom} ({\mathbb {H} }^{n})=O_{0}(n,1)}$

In the case of orientable manifolds, the monodrome display shows . ${\displaystyle \operatorname {Isom} ^{+}(H^{n})=SO_{0}(n,1)}$

literature

• Benedetti, Riccardo; Petronio, Carlo: Lectures on hyperbolic geometry. University text. Springer-Verlag, Berlin, 1992. xiv + 330 pp. ISBN 3-540-55534-X
• Kapovich, Michael: Hyperbolic manifolds and discrete groups. Reprint of the 2001 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2009. xxviii + 467 pp. ISBN 978-0-8176-4912-8

Web links

• John Milnor: Hyperbolic Geometry: The First 150 Years (PDF; 1.5 MB)