## Diffeomorphism Wikipedia

### Diffeomorphism - Wikipedia.

The diffeomorphism group of M is the group of all C r diffeomorphisms of M to itself, denoted by Diff r (M) or, when r is understood, Diff(M). This is a "large" group, in the sense that--provided M is not zero-dimensional--it is not locally compact. Topology. The diffeomorphism group has two natural topologies: weak and strong (Hirsch 1997)..

https://en.wikipedia.org/wiki/Diffeomorphism.

### Teichmüller space - Wikipedia.

The Fenchel-Nielsen coordinates (so named after Werner Fenchel and Jakob Nielsen) on the Teichmuller space () are associated to a pants decomposition of the surface .This is a decomposition of into pairs of pants, and to each curve in the decomposition is associated its length in the hyperbolic metric corresponding to the point in Teichmuller space, and another ....

https://en.wikipedia.org/wiki/Teichm%C3%BCller_space.

### General covariance - Wikipedia.

In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical laws under arbitrary differentiable coordinate transformations.The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the ....

https://en.wikipedia.org/wiki/General_covariance.

### Isomorphism - Wikipedia.

A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically differentiable manifolds. A permutation is an automorphism of a set. In geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations..

https://en.wikipedia.org/wiki/Isomorphism.

### Leibniz integral rule - Wikipedia.

This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and is particularly useful in the differentiation of integral transforms.An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable..

https://en.wikipedia.org/wiki/Leibniz_integral_rule.

### Lattice (group) - Wikipedia.

In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinatewise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point..

https://en.wikipedia.org/wiki/Lattice_(group).

### Wheeler–DeWitt equation - Wikipedia.

The Wheeler-DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt.The equation attempts to mathematically combine the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity.. In this approach, time plays a role different from what it does in non ....

https://en.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt_equation.

### Bifurcation theory - Wikipedia.

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter ....

https://en.wikipedia.org/wiki/Bifurcation_theory.

### Irreducible representation - Wikipedia.

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (,) or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation (|,), with closed under the action of {():}.. Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations..

https://en.wikipedia.org/wiki/Irreducible_representation.

### Edge of chaos - Wikipedia.

The edge of chaos is a transition space between order and disorder that is hypothesized to exist within a wide variety of systems. This transition zone is a region of bounded instability that engenders a constant dynamic interplay between order and disorder. Even though the idea of the edge of chaos is an abstract one, it has many applications in such fields as ecology, business ....

https://en.wikipedia.org/wiki/Edge_of_chaos.

### Hole argument - Wikipedia.

In general relativity, the hole argument is an apparent paradox that much troubled Albert Einstein while developing his famous field equations.. Some philosophers of physics take the argument to raise a problem for manifold substantialism, a doctrine that the manifold of events in spacetime is a "substance" which exists independently of the metric field defined on it or the matter within it..

https://en.wikipedia.org/wiki/Hole_argument.

### Dihedral group - Wikipedia.

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.. The notation for the dihedral group differs in geometry and abstract algebra.In geometry, D n or Dih n refers to the ....

https://en.wikipedia.org/wiki/Dihedral_group.

### Homogeneous space - Wikipedia.

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 are called the symmetries of X.A special case of this is when the group G in question is the automorphism group of the space X - here "automorphism ....

https://en.wikipedia.org/wiki/Homogeneous_space.

### G2 (mathematics) - Wikipedia.

In mathematics, G 2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras, as well as some algebraic groups.They are the smallest of the five exceptional simple Lie groups.G 2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.. The compact form of G 2 can be described as the ....

https://en.wikipedia.org/wiki/G2_(mathematics).

### Exponential map (Lie theory) - Wikipedia.

Definitions. Let be a Lie group and be its Lie algebra (thought of as the tangent space to the identity element of ).The exponential map is a map : which can be defined in several different ways. The typical modern definition is this: Definition: The exponential of is given by = where : is the unique one-parameter subgroup of whose tangent vector at the identity is equal to ..

https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory).

### Quotient group - Wikipedia.

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements ....

https://en.wikipedia.org/wiki/Quotient_group.

### Automorphism - Wikipedia.

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group.It is, loosely speaking, the symmetry group of the object..

https://en.wikipedia.org/wiki/Automorphism.

### Quaternion group - Wikipedia.

In group theory, the quaternion group Q 8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset {,,,,,} of the quaternions under multiplication. It is given by the group presentation = ?,,, ? =, = = = = ? , where e is the identity element and e commutes with the other elements of the group.. Another presentation of Q 8 is.

https://en.wikipedia.org/wiki/Quaternion_group.

### Connected sum - Wikipedia.

Connected sum at a point. A connected sum of two m-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres.. If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation.Although the construction uses the choice of the balls, the result ....

https://en.wikipedia.org/wiki/Connected_sum.