A Picard horn, also called the Picard topology or Picard model, is one of the oldest known

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

s, first described by

Émile Picard Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924. Life He was born in Paris on 24 July 1856 and educated there at t ...

in 1884. The manifold is the quotient of the upper half-plane model of hyperbolic 3-space by the

projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...

\operatorname_2(\mathbf

. It was proposed as a model for the shape of the universe in 2004. The term "horn" is due to

pseudosphere In geometry, a pseudosphere is a surface with constant negative Gaussian curvature. A pseudosphere of radius is a surface in \mathbb^3 having curvature in each point. Its name comes from the analogy with the sphere of radius , which is a surface ...

models of hyperbolic space.

Geometry and topology

A modern description, in terms of fundamental domain and identifications, can be found in section 3.2, page 63 of Grunewald and Huntebrinker, along with the first 80 eigenvalues of the

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...

, tabulated on page 72, where

\Upsilon_1

is a fundamental domain of the Picard space.Fritz Grunewald and Wolfgang Huntebrinker,
A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp
', Experiment. Math. Volume 5, Issue 1 (1996), 57-80

Cosmology

The term was coined in 2004 by Ralf Aurich, Sven Lustig, Frank Steiner, and Holger Then in their paper ''Hyperbolic Universes with a Horned Topology and the CMB Anisotropy''. The model was chosen in an attempt to describe the

microwave background radiation In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...

apparent in the universe, and has finite

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

and useful spectral characteristics (the first several eigenvalues of the Laplacian are computed and in good accord with observation). In this model one end of the figure curves finitely into the bell of the horn. The curve along any side of horn is considered to be a negative curve. The other end extends to infinity.

References

{{Manifolds 3-manifolds Hyperbolic geometry Physical cosmology

Geometry and topology

Cosmology

See also

References