In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Clifford–Klein form is a
double coset In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let be a group, and let and be subgroups. Let act on by left multi ...
space
:Γ\''G''/''H'',
where ''G'' is a
reductive Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
, ''H'' a closed subgroup of ''G'', and Γ a
discrete subgroup
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
of G that acts
properly discontinuously
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
on the
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
''G''/''H''. A suitable discrete subgroup Γ may or may not exist, for a given ''G'' and ''H''. If Γ exists, there is the question of whether Γ\''G''/''H'' can be taken to be a
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
, called a compact Clifford–Klein form.
When ''H'' is itself compact, classical results show that a compact Clifford–Klein form exists. Otherwise it may not, and there are a number of negative results.
History
According to
Moritz Epple
Moritz Epple (born 7 May 1960, in Stuttgart) is a German mathematician and historian of science.
Biography
Epple studied mathematics, philosophy and physics in Copenhagen, London, and at the University of Tübingen, where he received in 1987 his ...
, the Clifford-Klein forms began when
W. K. Clifford used
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s to ''twist'' their space. "Every twist possessed a space-filling family of invariant lines", the
Clifford parallel In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in s ...
s. They formed "a particular structure embedded in elliptic 3-space", the
Clifford surface, which demonstrated that "the same local geometry may be tied to spaces that are globally different."
Wilhelm Killing
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.
Life
Killing studied at the University of Mü ...
thought that for free mobility of rigid bodies there are four spaces: Euclidean, hyperbolic, elliptic and spherical. They are spaces of
constant curvature
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvatur ...
but constant curvature differs from free mobility: it is local, the other is both local and global. Killing's contribution to Clifford-Klein space forms involved formulation in terms of
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
s, finding new classes of examples, and consideration of the scientific relevance of spaces of constant curvature. He took up the task to develop physical theories of CK space forms.
Karl Schwarzchild wrote “The admissible measure of the curvature of space”, and noted in an appendix that physical space may actually be a non-standard space of constant curvature.
See also
*
Killing-Hopf theorem
*
Space form
Space is the boundless Three-dimensional space, three-dimensional extent in which Physical body, objects and events have relative position (geometry), position and direction (geometry), direction. In classical physics, physical space is often ...
References
*
Moritz Epple
Moritz Epple (born 7 May 1960, in Stuttgart) is a German mathematician and historian of science.
Biography
Epple studied mathematics, philosophy and physics in Copenhagen, London, and at the University of Tübingen, where he received in 1987 his ...
(2003
From Quaternions to Cosmology: Spaces of Constant Curvature ca. 1873 — 1925 invited address to International Congress of Mathematicians
*
*
{{DEFAULTSORT:Clifford-Klein Form
Lie groups
Homogeneous spaces