Кербездік метрикасы бар 3 өлшемді манифольдтар

Manifold of dimension 3 equipped with a hyperbolic metric
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 is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3 dimensional hyperbolic space by a discrete group of isometries (a Kleinian group). Hyperbolic 3 manifolds of finite volume have a particular importance in 3 dimensional topology as follows from Thurston's geometrisation conjecture proved by Perelman. The study of Kleinian groups is also an important topic in geometric group theory.

Hyperbolic geometry is the most rich and least understood of the eight geometries in dimension 3 (for example, for all other geometries it is not hard to give an explicit enumeration of the finite volume manifolds with this geometry, while this is far from being the case for hyperbolic manifolds). After the proof of the Geometrisation conjecture, understanding the topological properties of hyperbolic 3 manifolds is thus a major goal of 3 dimensional topology. Recent breakthroughs of Kahn–Markovic, Wise, Agol and others have answered most long standing open questions on the topic but there are still many less prominent ones which have not been solved. In dimension 2 almost all closed surfaces are hyperbolic (all but the sphere, projective plane, torus and Klein bottle). In dimension 3 this is far from true: there are many ways to construct infinitely many non hyperbolic closed manifolds. On the other hand, the heuristic statement that "a generic 3 manifold tends to be hyperbolic" is verified in many contexts. For example, any knot which is not either a satellite knot or a torus knot is hyperbolic. Moreover, almost all Dehn surgeries on a hyperbolic knot yield a hyperbolic manifold. A similar result is true of links (Thurston's hyperbolic Dehn surgery theorem), and since all 3 manifolds are obtained as surgeries on a link in the 3 sphere this gives a more precise sense to the informal statement. Another sense in which "almost all" manifolds are hyperbolic in dimension 3 is that of random models. For example, random Heegaard splittings of genus at least 2 are almost surely hyperbolic (when the complexity of the gluing map goes to infinity). The relevance of the hyperbolic geometry of a 3 manifold to its topology also comes from the Mostow rigidity theorem, which states that the hyperbolic structure of a hyperbolic 3 manifold of finite volume is uniquely determined by its homotopy type. In particular geometric invariant such as the volume can be used to define new topological invariants.

The thick thin decomposition is valid for all hyperbolic 3 manifolds, though in general the thin part is not as described above. A hyperbolic 3 manifold is said to be geometrically finite if it contains a convex submanifold (its convex core) onto which it retracts, and whose thick part is compact (note that all manifolds have a convex core, but in general it is not compact). The simplest case is when the manifold does not have "cusps" (i. e. the fundamental group does not contain parabolic elements), in which case the manifold is geometrically finite if and only if it is the quotient of a closed, convex subset of hyperbolic space by a group acting cocompactly on this subset.

In the previous construction the manifolds obtained are always compact. To obtain manifolds with cusps one has to use polytopes which have ideal vertices (i. e. vertices which lie on the sphere at infinity). In this setting the gluing construction does not always yield a complete manifold. Completeness is detected by a system of equations involving the dihedral angles around the edges adjacent to an ideal vertex, which are commonly called Thurston's gluing equations. In case the gluing is complete the ideal vertices become cusps in the manifold. An example of a noncompact, finite volume hyperbolic manifold obtained in this way is the Gieseking manifold which is constructed by gluing faces of a regular ideal hyperbolic tetrahedron together. It is also possible to construct a finite volume, complete hyperbolic manifold when the gluing is not complete. In this case the completion of the metric space obtained is a manifold with a torus boundary and under some (not generic) conditions it is possible to glue a hyperbolic solid torus on each boundary component so that the resulting space has a complete hyperbolic metric. Topologically, the manifold is obtained by hyperbolic Dehn surgery on the complete hyperbolic manifold which would result from a complete gluing. It is not known whether all hyperbolic 3 manifolds of finite volume can be constructed in this way. In practice however this is how computational software (such as SnapPea or Regina) stores hyperbolic manifolds.

The construction of arithmetic Kleinian groups from quaternion algebras gives rise to particularly interesting hyperbolic manifolds. On the other hand, they are in some sense "rare" among hyperbolic 3 manifolds (for example hyperbolic Dehn surgery on a fixed manifold results in a non arithmetic manifold for almost all parameters).

In contrast to the explicit constructions above it is possible to deduce the existence of a complete hyperbolic structure on a 3 manifold purely from topological information. This is a consequence of the Geometrisation conjecture and can be stated as follows (a statement sometimes referred to as the "hyperbolisation theorem", which was proven by Thurston in the special case of Haken manifolds):

If a compact 3 manifold with toric boundary is irreducible and algebraically atoroidal (meaning that every injectively immersed torus is homotopic to a boundary component) then its interior carries a complete hyperbolic metric of finite volume. A particular case is that of a surface bundle over the circle: such manifolds are always irreducible, and they carry a complete hyperbolic metric if and only if the monodromy is a pseudo Anosov map. Another consequence of the Geometrisation conjecture is that any closed 3 manifold which admits a Riemannian metric with negative sectional curvatures admits in fact a Riemannian metric with constant sectional curvature 1. This is not true in higher dimensions.

A sequence of Kleinian groups is said to be geometrically convergent if it converges in the Chabauty topology. For the manifolds obtained as quotients this amounts to them being convergent in the pointed Gromov Hausdorff metric.

The hyperbolic volume can be used to order the space of all hyperbolic manifold. The set of manifolds corresponding to a given volume is at most finite, and the set of volumes is well ordered and of order type More precisely, Thurston's hyperbolic Dehn surgery theorem implies that a manifold with cusps is a limit of a sequence of manifolds with cusps for any , so that the isolated points are volumes of compact manifolds, the manifolds with exactly one cusp are limits of compact manifolds, and so on. Together with results of Jørgensen the theorem also proves that any convergent sequence must be obtained by Dehn surgeries on the limit manifold.

Sequences of quasi fuchsian surface groups of given genus can converge to a doubly degenerate surface group, as in the double limit theorem.