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91 relations: Abelian group, Alexander's trick, Allen Hatcher, Angle, Étale morphism, Baire space, Banach manifold, Banach space, Bijection, Brouwer fixed-point theorem, Compactification (mathematics), Complex number, Configuration space (physics), Continuous function, Deformation (mechanics), Dehn twist, Derivative, Differentiable manifold, Differential of a function, Dimension, Dual number, Exotic R4, Exotic sphere, Exponential map (Riemannian geometry), Fiber bundle, Fréchet manifold, Fréchet space, Fundamental group, General linear group, Gustave Choquet, Hausdorff space, Hellmuth Kneser, Homeomorphism, Hyperbolic angle, Identity component, Immersion (mathematics), Inverse function, Invertible matrix, Isomorphism, Jacobian matrix and determinant, Jakob Nielsen (mathematician), John Milnor, Large diffeomorphism, Lie algebra, Lie bracket of vector fields, Lie group, Linear map, Local diffeomorphism, Manifold, Map (mathematics), ..., Mapping class group, Mathematics, Max Dehn, Mechanics, Michael Freedman, Modular group, Neighbourhood (mathematics), Nielsen–Thurston classification, Orthogonal group, Outer automorphism group, Poisson kernel, Presentation of a group, Proper map, Pseudo-Anosov map, Pushforward (differential), Quotient group, René Thom, Retract, Riemannian manifold, Second-countable space, Simon Donaldson, Simply connected space, Slope, Smoothness, Split-complex number, Springer Science+Business Media, Stephen Smale, Submersion (mathematics), Supermanifold, Surface (topology), Teichmüller space, Tibor Radó, Topology (journal), Torus, Unit circle, Unit disk, Vector field, W. B. R. Lickorish, William Thurston, 3-sphere, 4-manifold. Expand index (41 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Alexander's trick
Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.
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Allen Hatcher
Allen Edward Hatcher (born October 23, 1944) is an American topologist.
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Angle
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
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Étale morphism
In algebraic geometry, an étale morphism is a morphism of schemes that is formally étale and locally of finite presentation.
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Baire space
In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense.
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Banach manifold
In mathematics, a Banach manifold is a manifold modeled on Banach spaces.
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Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
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Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.
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Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer.
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Compactification (mathematics)
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Configuration space (physics)
In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system.
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Deformation (mechanics)
Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration.
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Dehn twist
In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).
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Derivative
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).
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Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
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Differential of a function
In calculus, the differential represents the principal part of the change in a function y.
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Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
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Dual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2.
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Exotic R4
In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R4.
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Exotic sphere
In differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere.
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Exponential map (Riemannian geometry)
In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself.
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Fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.
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Fréchet manifold
In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.
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Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
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Fundamental group
In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.
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General linear group
In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.
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Gustave Choquet
Gustave Choquet (1 March 1915 – 14 November 2006) was a French mathematician.
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Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.
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Hellmuth Kneser
Hellmuth Kneser (16 April 1898 – 23 August 1973) was a Baltic German mathematician, who made notable contributions to group theory and topology.
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Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
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Hyperbolic angle
In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle. The hyperbolic angle is first defined for a "standard position", and subsequently as a measure of an interval on a branch of a hyperbola. A hyperbolic angle in standard position is the angle at (0, 0) between the ray to (1, 1) and the ray to (x, 1/x) where x > 1. The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is ln x. Note that unlike circular angle, hyperbolic angle is unbounded, as is the function ln x, a fact related to the unbounded nature of the harmonic series. The hyperbolic angle in standard position is considered to be negative when 0 a > 1 so that (a, b) and (c, d) determine an interval on the hyperbola xy.
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Identity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group.
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Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.
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Inverse function
In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.
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Invertible matrix
In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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Jacobian matrix and determinant
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
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Jakob Nielsen (mathematician)
Jakob Nielsen (15 October 1890 in Mjels, Als – 3 August 1959 in Helsingør) was a Danish mathematician known for his work on automorphisms of surfaces.
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John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, K-theory and dynamical systems.
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Large diffeomorphism
In mathematics and theoretical physics, a large diffeomorphism is an equivalence class of diffeomorphisms under the equivalence relation where diffeomorphisms that can be continuously connected to each other are in the same equivalence class.
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Lie algebra
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
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Lie bracket of vector fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Linear map
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
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Local diffeomorphism
In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a function between smooth manifolds that preserves the local differentiable structure.
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Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
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Map (mathematics)
In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.
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Mapping class group
In mathematics, in the sub-field of geometric topology, the mapping class group is an important algebraic invariant of a topological space.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German-born American mathematician and student of David Hilbert.
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Mechanics
Mechanics (Greek μηχανική) is that area of science concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.
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Michael Freedman
Michael Hartley Freedman (born 21 April 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara.
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Modular group
In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.
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Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
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Nielsen–Thurston classification
In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface.
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Orthogonal group
In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.
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Outer automorphism group
In mathematics, the outer automorphism group of a group,, is the quotient,, where is the automorphism group of and) is the subgroup consisting of inner automorphisms.
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Poisson kernel
In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc.
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Presentation of a group
In mathematics, one method of defining a group is by a presentation.
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Proper map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact.
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Pseudo-Anosov map
In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface.
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Pushforward (differential)
Suppose that is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. It can be viewed as a generalization of the total derivative of ordinary calculus.
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Quotient group
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 the group structure.
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René Thom
René Frédéric Thom (2 September 1923 – 25 October 2002) was a French mathematician.
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Retract
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace which preserves the position of all points in that subspace.
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Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
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Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.
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Simon Donaldson
Sir Simon Kirwan Donaldson FRS (born 20 August 1957), is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson–Thomas theory.
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Simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.
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Slope
In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.
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Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.
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Split-complex number
In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written z.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan.
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Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.
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Supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry.
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Surface (topology)
In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.
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Teichmüller space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism.
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Tibor Radó
Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I.
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Topology (journal)
Topology was a peer-reviewed mathematical journal covering topology and geometry.
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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
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Unit circle
In mathematics, a unit circle is a circle with a radius of one.
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Unit disk
In mathematics, the open unit disk (or disc) around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1: The closed unit disk around P is the set of points whose distance from P is less than or equal to one: Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself.
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Vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
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W. B. R. Lickorish
William Bernard Raymond Lickorish (born 19 February 1938) is a mathematician.
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William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.
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3-sphere
In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.
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4-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold.
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References
[1] https://en.wikipedia.org/wiki/Diffeomorphism
