In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invaria
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 homo
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian prod
In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used
Group actions in computational anatomy
Group actions are central to Riemannian geometry and defining orbits (control theory). The orbits of computational anatomy consist of anatomical shapes and medical images; the anatomical shapes are su
In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H by automorphisms (which we will write on the left, , and a homomorphism of groups that
Quotient space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the or
In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithful
In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common genera
Continuous group action
In topology, a continuous group action on a topological space X is a group action of a topological group G that is continuous: i.e., is a continuous map. Together with the group action, X is called a
Lie group action
In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable.
In the mathematics of moduli theory, given an algebraic, reductive, Lie group and a finitely generated group , the -character variety of is a space of equivalence classes of group homomorphisms from t
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and cod
Cocompact group action
In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space. If X is locally compact, then an equivalent condition is that there is a comp
In mathematics, a CAT(k) group is a group that acts discretely, cocompactly and isometrically on a CAT(k) space.
In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phas
Group with operators
In abstract algebra, a branch of mathematics, the algebraic structure group with operators or Ω-group can be viewed as a group with a set Ω that operates on the elements of the group in a special way.
In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps , and while equivariant top
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordina
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space
Borel fixed-point theorem
In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem. The result was proved by Armand Borel.
Principal orbit type theorem
In mathematics, the principal orbit type theorem states that compact Lie group acting smoothly on a connected differentiable manifold has a principal orbit type.