In
mathematics, a hyperbolic manifold is a space where every point looks locally like
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
of some dimension. They are especially studied in dimensions 2 and 3, where they are called
hyperbolic surfaces and
hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It ...
s, respectively. In these dimensions, they are important because most
manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
can be made into a hyperbolic manifold by a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
. This is a consequence of the
uniformization theorem
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
for surfaces and the
geometrization theorem for 3-manifolds proved by
Perelman
Perelman ( he, פרלמן) is an Ashkenazi Jewish surname. Notable people with the surname include:
* Bob Perelman (b. 1947), American poet
* Chaim Perelman (1912-1984), Polish-born Belgian philosopher of law
* Eliezer Ben-Yehuda () (1858-1922), ...
.
Rigorous Definition
A hyperbolic
-manifold is a complete
Riemannian -manifold of constant
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...
.
Every complete, connected, simply-connected manifold of constant negative curvature
is
isometric to the real hyperbolic space
. As a result, the universal cover of any closed manifold
of constant negative curvature
is
. Thus, every such
can be written as
where
is a torsion-free discrete group of isometries on
. That is,
is a discrete subgroup of
. The manifold has finite volume if and only if
is a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
.
Its
thick–thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and ends which are the product of a Euclidean (
)-manifold and the closed half-ray. The manifold is of finite volume if and only if its thick part is compact.
Examples
The simplest example of a hyperbolic manifold is
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
, as each point in hyperbolic space has a neighborhood isometric to hyperbolic space.
A simple non-trivial example, however, is the
once-punctured torus. This is an example of an
(Isom(), )-manifold. This can be formed by taking an ideal rectangle in
– that is, a rectangle where the vertices are on the boundary at infinity, and thus don't exist in the resulting manifold – and identifying opposite images.
In a similar fashion, we can construct the thrice-punctured sphere, shown below, by gluing two ideal triangles together. This also shows how to draw curves on the surface – the black line in the diagram becomes the closed curve when the green edges are glued together. As we are working with a punctured sphere, the colored circles in the surface – including their boundaries – are not part of the surface, and hence are represented in the diagram as
ideal vertices.
Many
knots and links, including some of the simpler knots such as the
figure eight knot and the
Borromean rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the t ...
, are hyperbolic, and so the complement of the knot or link in
is a hyperbolic 3-manifold of finite volume.
Important Results
For
the hyperbolic structure on a ''finite volume'' hyperbolic
-manifold is unique by
Mostow rigidity Mostow may refer to: People
* George Mostow (1923–2017), American mathematician
** Mostow rigidity theorem
* Jonathan Mostow
Jonathan Mostow (born November 28, 1961) is an American film director, screenwriter, and producer. He has directed f ...
and so geometric invariants are in fact topological invariants. One of these geometric invariants used as a topological invariant is the
hyperbolic volume of a knot or link complement, which can allow us to distinguish two knots from each other by studying the geometry of their respective manifolds.
We can also ask what the area of the boundary of the knot complement is. As there is a relationship between the volume of a knot complement and the volume of the complement under
Dehn filling,
we can use the area of the boundary to inform us of how the volume might change under such a filling.
See also
*
Hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It ...
*
Hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
*
Hyperbolization theorem
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.
Statement
One form of Thurston's geometrization theor ...
*
Margulis lemma
*
Normally hyperbolic invariant manifold A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a hyperbolic fixed point and a hyperbolic set. The difference can be described heuristically as follows: For a manifold \Lambda to be normally hyperbolic we are al ...
References
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{{Manifolds
Hyperbolic geometry
Manifolds
Riemannian manifolds