Hyperbolic space in the context of "Riemannian metric"

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.

A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the packing density of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume.

In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object along with the object itself. For example, a 1-dimensional line ${\displaystyle (l)}$ may be studied in isolation —in which case the ambient space of ${\displaystyle l}$ is ${\displaystyle l}$, or it may be studied as an object embedded in 2-dimensional Euclidean space ${\displaystyle (\mathbb {R} ^{2})}$—in which case the ambient space of ${\displaystyle l}$ is ${\displaystyle \mathbb {R} ^{2}}$, or as an object embedded in 2-dimensional hyperbolic space ${\displaystyle (\mathbb {H} ^{2})}$—in which case the ambient space of ${\displaystyle l}$ is ${\displaystyle \mathbb {H} ^{2}}$. To see why this makes a difference, consider the statement "Parallel lines never intersect." This is true if the ambient space is ${\displaystyle \mathbb {R} ^{2}}$, but false if the ambient space is ${\displaystyle \mathbb {H} ^{2}}$, because the geometric properties of ${\displaystyle \mathbb {R} ^{2}}$ are different from the geometric properties of ${\displaystyle \mathbb {H} ^{2}}$. All spaces are subsets of their ambient space.

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.

Johannes Trolle Hjelmslev (Danish: [ˈjelˀmsle̝w]; 7 April 1873 – 16 February 1950) was a mathematician from Hørning, Denmark. Hjelmslev worked in geometry and history of geometry. He was the discoverer and eponym of the Hjelmslev transformation, a method for mapping an entire hyperbolic plane into a circle with a finite radius.He was the father of Louis Hjelmslev.

Originally named Johannes Trolle Petersen, he changed his patronymic to the surname Hjelmslev to avoid confusion with Julius Petersen. Some of his results are known under his original name, including the Petersen–Morley theorem.

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.