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Isomorphism of differentiable manifolds
A C 1 {\displaystyle C^{1}} -diffeomorphism is simply a diffeomorphism, and a C 0 {\displaystyle C^{0}} -diffeomorphism is a homeomorphism. Given a subset
Diffeomorphism
Diffeomorphism that has a hyperbolic structure on the tangent bundle
Bernoulli map, and Arnold's cat map. If the map is a diffeomorphism, then it is called an Anosov diffeomorphism. If a flow on a manifold splits the tangent bundle
Anosov_diffeomorphism
Smooth map which is a diffeomorphism upon restriction
U → f ( U ) {\displaystyle f\vert _{U}:U\to f(U)} is a diffeomorphism. A local diffeomorphism is a special case of an immersion f : X → Y {\displaystyle
Local_diffeomorphism
Branch of mathematics
of all smooth manifolds up to diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often
Differential_topology
Definition of a class of dynamical systems
Anosov system. Let M be a smooth manifold with a diffeomorphism f: M→M. Then f is an axiom A diffeomorphism if the following two conditions hold: The nonwandering
Axiom_A
Constraint in diffeomorphism invariant theories
theoretical physics, it is often important to study theories with the diffeomorphism symmetry such as general relativity. These theories are invariant under
Diffeomorphism_constraint
Type of diffeomorphism or homeomorphism of a surface
pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its definition
Pseudo-Anosov_map
Topics referred to by the same term
morphisms (or both) Large diffeomorphism, a diffeomorphism that cannot be continuously connected to the identity diffeomorphism in mathematics and physics
Large
Principle stating that physical laws are the same in all coordinate systems
In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical
General_covariance
Representation theory of the symmetries of manifolds
orientation-preserving diffeomorphism group of M (only the identity component of mappings homotopic to the identity diffeomorphism if you wish) and Diffx1(M)
Representation theory of diffeomorphism groups
Representation_theory_of_diffeomorphism_groups
Application of differential geometry
imaging. The study of deformable shapes in CA rely on high-dimensional diffeomorphism groups Diff V {\displaystyle \operatorname {Diff} _{V}} which generate
Riemannian metric and Lie bracket in computational anatomy
Riemannian_metric_and_Lie_bracket_in_computational_anatomy
On extending a Lie group action on a manifold to an equivariant diffeomorphism
G\times _{G_{x}}T_{x}M/T_{x}(G\cdot x)} so that it defines an equivariant diffeomorphism from the neighborhood to its image, which contains the orbit of x {\displaystyle
Slice theorem (differential geometry)
Slice_theorem_(differential_geometry)
Field equation from quantum gravity
commutation relations with the diffeomorphism constraints generate the Bergman–Komar "group" (which is the diffeomorphism group on-shell). In canonical
Wheeler–DeWitt_equation
Topological space in group theory
group elements are diffeomorphisms. The structure of a G-space is a group homomorphism ρ : G → Diffeo(X) into the diffeomorphism group of X. Riemannian
Homogeneous_space
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
structures on the 7-sphere. In any dimension Milnor (1959) showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of
Exotic_sphere
Class of diffeomorphism
theoretical physics, a large diffeomorphism is an equivalence class of diffeomorphisms under the equivalence relation where diffeomorphisms that can be continuously
Large_diffeomorphism
Mathematical operation
ϕ {\displaystyle \phi } . When the map ϕ {\displaystyle \phi } is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform
Pullback (differential geometry)
Pullback_(differential_geometry)
Group that is also a differentiable manifold with group operations that are smooth
whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Observe that, by the above, a continuous
Lie_group
Breakdown of general covariance at the quantum level
synonymous with diffeomorphism anomaly, since general covariance is symmetry under coordinate reparametrization; i.e. diffeomorphism. General covariance
Gravitational_anomaly
Metric study of shape and form in computational anatomy
The study of images in computational anatomy rely on high-dimensional diffeomorphism groups φ ∈ Diff V {\displaystyle \varphi \in \operatorname {Diff} _{V}}
Diffeomorphometry
American mathematician
states: Let f be a diffeomorphism of a compact manifold with a nonwandering point x. Then, there is (in the space of diffeomorphisms, equipped with the
Charles_C._Pugh
Theory of quantum gravity merging quantum mechanics and general relativity
spatial diffeomorphism on γ {\displaystyle \gamma } instead. Therefore, the meaning of O ^ ′ {\displaystyle {\hat {O}}'} is a spatial diffeomorphism on γ
Loop_quantum_gravity
is a collar neighbourhood of M {\displaystyle M} whenever there is a diffeomorphism f : ∂ M × [ 0 , 1 ) → U {\displaystyle f:\partial M\times [0,1)\to U}
Collar_neighbourhood
Interdisciplinary field of biology
more general diffeomorphism group has been the group of choice, which is the infinite dimensional analogue. The high-dimensional diffeomorphism groups used
Computational_anatomy
Nonlinear differential operator used to study conformal mappings
interpreted as a continuous 1-cocycle or crossed homomorphism of the diffeomorphism group of the circle with coefficients in the module of densities of
Schwarzian_derivative
Image processing step or image registration method
transformations homeomorphisms and diffeomorphisms since they carry smooth submanifolds smoothly during transformation. Diffeomorphisms are generated in the modern
Spatial_normalization
Theorem that the diffeomorphism group of the 3-sphere has the homotopy-type of O(4)
Smale conjecture, named after Stephen Smale, is the statement that the diffeomorphism group of the 3-sphere has the homotopy-type of its isometry group, the
Smale_conjecture
Topics referred to by the same term
In mathematics, Denjoy's theorem may refer to several theorems proved by Arnaud Denjoy, including Denjoy–Carleman theorem Denjoy–Koksma inequality Denjoy–Luzin
Denjoy_theorem
American mathematician (born 1930)
awarded the Wolf Prize in mathematics. Smale proved that the oriented diffeomorphism group of the two-dimensional sphere has the same homotopy type as the
Stephen_Smale
When a diffeomorphism of the circle is topologically conjugate to an irrational rotation
gives a sufficient condition for a diffeomorphism of the circle to be topologically conjugate to a diffeomorphism of a special kind, namely an irrational
Denjoy's theorem on rotation number
Denjoy's_theorem_on_rotation_number
Algebraic geometry
étale if it has a lifting property that is analogous to being a local diffeomorphism. Let A be a topological ring, and let B be a topological A-algebra.
Formally_étale_morphism
Monster and modular connection
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Monstrous_moonshine
Extends the Jordan curve theorem to characterize the inner and outer regions
differ by a diffeomorphism of the unit circle. On the other hand, a diffeomorphism f of the unit circle can be extended to a diffeomorphism F of the unit
Schoenflies_problem
the entire manifold M is hyperbolic, the map f is called an Anosov diffeomorphism. The dynamics of f on a hyperbolic set, or hyperbolic dynamics, exhibits
Hyperbolic_set
a Hamiltonian function H). A Hamiltonian diffeomorphism of a symplectic manifold (M, ω) is a diffeomorphism Φ of M which is the integral of a smooth path
Spectral_invariants
Rwandan-born American mathematician (born 1947)
no. 3, 215–229. MR 0561971 Augustin Banyaga, On Isomorphic Classical Diffeomorphism Groups. I., Proceedings of the American Mathematical Society 98 (1986)
Augustin_Banyaga
unique harmonic function u : D → Ω such that u = μ on ∂D and u is a diffeomorphism. R. Schoen, S. T. Yau. (1997) Lectures on Harmonic Maps. International
Radó's theorem (harmonic functions)
Radó's_theorem_(harmonic_functions)
Point on a curve where motion must move backwards
differentiable functions: a curve has a cusp at a point if there is a diffeomorphism of a neighborhood of the point in the ambient space, which maps the
Cusp_(singularity)
Group of 𝑛 × 𝑛 invertible matrices
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
General_linear_group
Mathematical conjecture
Hamiltonian diffeomorphism of M {\displaystyle M} . The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of M
Arnold_conjecture
Distance-preserving mathematical transformation
manifold to the metric tensor on the first. When such a map is also a diffeomorphism, such a map is called an isometry (or isometric isomorphism), and provides
Isometry
Group of unitary complex matrices with determinant of 1
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Special_unitary_group
Physical theory with fields invariant under the action of local "gauge" Lie groups
system can be chosen freely under arbitrary diffeomorphisms of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description
Gauge_theory
Key constraint in some theories admitting Hamiltonian formulations
constraint technically refers to a linear combination of spatial and time diffeomorphism constraints reflecting the reparametrizability of the theory under both
Hamiltonian_constraint
Parametrizes complex structures on a surface
isotopic to a holomorphic diffeomorphism. Such a pair is called a marked Riemann surface; the marking being the diffeomorphism; another definition of markings
Teichmüller_space
26-dimensional string theory
{\displaystyle T={\frac {1}{2\pi \alpha '}}} . I 0 {\displaystyle I_{0}} has diffeomorphism and Weyl invariance. Weyl symmetry is broken upon quantization (Conformal
Bosonic_string_theory
Way to join two given mathematical manifolds together
then the result is unique up to diffeomorphism. There are subtle problems in the smooth case: not every diffeomorphism between the boundaries of the spheres
Connected_sum
In mathematics, invertible homomorphism
spaces. A homeomorphism is an isomorphism of topological spaces. A diffeomorphism is an isomorphism of spaces equipped with a differential structure,
Isomorphism
Low-energy particles on event horizons
arises on a spinning black hole with or without charge due to a type of diffeomorphism called a "hidden conformal symmetry". This symmetry arises only when
Soft_hair_(black_holes)
Isomorphism of symplectic manifolds
structure of phase space, and is called a canonical transformation. A diffeomorphism between two symplectic manifolds f : ( M , ω ) → ( N , ω ′ ) {\displaystyle
Symplectomorphism
18 mathematical problems stated in 1998
three-sphere a minimal set (Gottschalk's conjecture)? Is an Anosov diffeomorphism of a compact manifold topologically the same as the Lie group model
Smale's_problems
Manifold that "locally looks like" Euclidean space
In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally
Flat_manifold
Concept in cosmology
troublesome contributions simply do not gravitate. Recently, a fully diffeomorphism-invariant action principle that gives the equations of motion for trace-free
Cosmological_constant_problem
Formulation of general relativity
The first class constraints of general relativity are the spatial diffeomorphism constraint and the Hamiltonian constraint (also known as the Wheeler–De
Canonical_quantum_gravity
Suite of algorithms
a variational problem in which the template is transformed via the diffeomorphism used as a change of coordinate to minimize a squared-error matching
Large deformation diffeomorphic metric mapping
Large_deformation_diffeomorphic_metric_mapping
Type of group and algebra representation
Representation theory of the Galilean group Representation theory of diffeomorphism groups Representation theory of the Poincaré group Theorem of the highest
Irreducible_representation
Two-dimensional manifold
higher-dimensional manifolds.) Thus closed surfaces are classified up to diffeomorphism by their Euler characteristic and orientability. Smooth surfaces equipped
Surface_(topology)
Concept in mathematics
category: if we replace all instances of "homeomorphism" above with "diffeomorphism" we obtain the same group, that is the inclusion Diff + ( S ) ⊂ Homeo
Mapping class group of a surface
Mapping_class_group_of_a_surface
differential geometry—a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics". More precisely, given two
Geodesic_map
Bijection of a set using properties of shapes in space
refined. Conformal transformation Equiareal transformation Homeomorphism Diffeomorphism Transformations of the same type form groups that may be sub-groups
Geometric_transformation
Conditions under which a chaotic system can be reconstructed by observation
space with k > 2 d A . {\displaystyle k>2d_{A}.} That is, there is a diffeomorphism φ that maps A into R k {\displaystyle \mathbb {R} ^{k}} such that the
Takens's_theorem
On the connectivity of a group of diffeomorphisms of a manifold
diffeomorphisms of a manifold. Given a differentiable manifold M (with or without boundary), a pseudo-isotopy diffeomorphism of M is a diffeomorphism
Pseudoisotopy_theorem
Differential geometry technique
up to a diffeomorphism. For example, if M and N are two Riemannian manifolds with metrics g and h, respectively, when is there a diffeomorphism ϕ : M →
Cartan's_equivalence_method
Map from a Lie algebra to its Lie group
diffeomorphism at all points. For example, the exponential map from s o {\displaystyle {\mathfrak {so}}} (3) to SO(3) is not a local diffeomorphism;
Exponential_map_(Lie_theory)
Mathematical function between groups that preserves multiplication structure
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Group_homomorphism
Group with subnormal series where all factors are abelian
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Solvable_group
Class of chaotic maps
to a diffeomorphism, the extension cannot always be done in the plane. For example, the map on the right needs to be extended to a diffeomorphism of the
Horseshoe_map
Group of unitary matrices
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Unitary_group
Branch of mathematics that studies the properties of groups
and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous. Most groups considered
Group_theory
Smooth manifold with an inner product on each tangent space
and ( N , h ) {\displaystyle (N,h)} are two Riemannian manifolds, a diffeomorphism f : M → N {\displaystyle f:M\to N} is called an isometry if g = f ∗
Riemannian_manifold
Root system associated to a symmetric space
F4 E6 E7 E8 Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean Lie algebras Lie group–Lie algebra correspondence Exponential
Restricted_root_system
Index of articles associated with the same name
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Janko_group
American mathematician (1942–2017)
"Commutators of diffeomorphisms." Commentarii Mathematici Helvetici 49.1 (1974): 512-528. Mather, John N. "Commutators of diffeomorphisms: II." Commentarii
John_N._Mather
Concept in mathematics
structural stability of diffeomorphisms of the circle. As a consequence of the Denjoy theorem, an orientation preserving C2 diffeomorphism ƒ of the circle is
Structural_stability
Tangent spaces of a manifold
an open contractible subset of M {\displaystyle M} , then there is a diffeomorphism T U → U × R n {\displaystyle TU\to U\times \mathbb {R} ^{n}} which restricts
Tangent_bundle
Commutative group (mathematics)
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Abelian_group
Trick relating differential forms
_{0}} and α 1 {\displaystyle \alpha _{1}} on a smooth manifold by a diffeomorphism ψ ∈ D i f f ( M ) {\displaystyle \psi \in \mathrm {Diff} (M)} such that
Moser's_trick
If G is a finitely generated group with exponent n, is G necessarily finite?
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Burnside_problem
Algebraic structure used in analysis
Lie algebra of the diffeomorphism group of X. So the Lie bracket of vector fields describes the non-commutativity of the diffeomorphism group. An action
Lie_algebra
Feature of systems that defy description
Limit set Lyapunov exponent Orbit Periodic point Phase space Anosov diffeomorphism Arnold tongue axiom A dynamical system Bifurcation diagram Box-counting
Complexity
Operation in differential geometry
neighborhood U of p. Abusing notation slightly, we may regard (xi) as a local diffeomorphism ( x i ) : M → R n {\displaystyle (x^{i}):M\rightarrow \mathbb {R} ^{n}}
Jet_(mathematics)
Representation theory of the symmetries of non-relativistic quantum space
classification Pauli–Lubanski pseudovector Representation theory of the diffeomorphism group Rotation operator Bargmann, V. (1954). "On Unitary Ray Representations
Representation theory of the Galilean group
Representation_theory_of_the_Galilean_group
Analysis of geometric properties
investigating deformations transforming one shape into another. In particular a diffeomorphism preserves smoothness in the deformation. This was pioneered in D'Arcy
Statistical_shape_analysis
Mathematical technique for simplification
{\displaystyle \Phi :A\rightarrow B} be a C r {\displaystyle C^{r}} -diffeomorphism between them, that is: Φ {\displaystyle \Phi } is a r {\displaystyle
Change_of_variables
Equivalence class of spin networks
gravitational field. Living Reviews in Relativity: Loop Quantum Gravity: Diffeomorphism invariance Rovelli, Carlo (1996-10-14). "Black Hole Entropy from Loop
S-knot
Condition in which spacetime itself breaks down
conical singularity occurs when there is a point where the limit of some diffeomorphism invariant quantity does not exist or is infinite, in which case spacetime
Gravitational_singularity
Theory of gravity in which the graviton has nonzero mass
Lagrangian for h μ ν {\displaystyle h_{\mu \nu }} that is consistent with diffeomorphism invariance, as well as a coupling to matter of the form h μ ν T μ ν
Massive_gravity
Mathematical behavior near singularities
consider its induced diffeomorphism on local transversal sections through the endpoints. Within a simply connected chart this diffeomorphism becomes unique
Monodromy
Type of subgroup of an algebraic group
F4 E6 E7 E8 Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean Lie algebras Lie group–Lie algebra correspondence Exponential
Borel_subgroup
Mapping which preserves all topological properties of a given space
Local homeomorphism – Mathematical function revertible near each point Diffeomorphism – Isomorphism of differentiable manifolds Uniform isomorphism – Uniformly
Homeomorphism
Vector field
vector field on M. Then X generates a one-parameter group of local diffeomorphisms FlXt, the flow along X. The differential of FlXt gives, for each t
Variational_vector_field
24 mathematical problems stated in 1982
software for calculation of canonical form of surface diffeomorphisms and group action of diffeomorphisms of projectivized lamination spaces. Addressed through
Thurston's_24_questions
Periodic set of points
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Lattice_(group)
Commutative group in which all nonzero elements have the same order
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Elementary_abelian_group
Key result in general relativity
Suppose that K is an open precompact subset of M such that there is a diffeomorphism Φ : ℝ3 − B1(0) → M − K, and suppose that there is a number m such that
Positive_energy_theorem
Subgroup invariant under conjugation
unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups
Normal_subgroup
Poisson integrals of homeomorphisms are diffeomorphisms
orientation preserving diffeomorphism of the open unit disk. To prove that Ff is locally an orientation-preserving diffeomorphism, it suffices to show that
Radó–Kneser–Choquet_theorem
Linear approximation of smooth maps on tangent spaces
not be invertible. However, if φ {\displaystyle \varphi } is a local diffeomorphism, then d φ x {\displaystyle d\varphi _{x}} is invertible, and the inverse
Pushforward_(differential)
Generalized matrix decomposition for Lie groups and Lie algebras
p ( X ) {\displaystyle (k,X)\mapsto k\cdot \mathrm {exp} (X)} is a diffeomorphism. The subgroup K {\displaystyle K} is a maximal compact subgroup of G
Cartan_decomposition
Theory of gravity
implementation of spacetime diffeomorphism invariance, but as an implementation of spatial relationalism based on spatial diffeomorphisms and spatial Weyl symmetry
Shape_dynamics
DIFFEOMORPHISM
DIFFEOMORPHISM
DIFFEOMORPHISM
DIFFEOMORPHISM
Girl/Female
Hindu, Indian
Beauty; Fish of God
Boy/Male
Hindu, Indian
Great; The Name of a Hindu God in India
Male
German
A derivative of German Reginar, RAINER means "wise warrior."
Boy/Male
Tamil
Thought, Devotion, Another name of the Sun, Lord Shiva
Girl/Female
Australian, Hebrew
Pearl
Boy/Male
Indian
More clear
Boy/Male
Hindu, Indian
Ruler; Like King
Boy/Male
Arabic, Muslim
A Person who Cuts off; Uproots; Name of a Sahabi who Participated in the Battle of Badr
Boy/Male
Indian, Sikh
Good
Male
English
(×ֲמַרְיָה) Anglicized form of Hebrew Amaryah, AMARIAH means "whom God spoke of." In the bible, this is the name of a priest who lived in the time of King Jehoshaphat.
DIFFEOMORPHISM
DIFFEOMORPHISM
DIFFEOMORPHISM
DIFFEOMORPHISM
DIFFEOMORPHISM