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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 diffeomorphism is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
of
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s. It is an
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
that maps one
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
to another such that both the function and its inverse are
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
.


Definition

Given two
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Ne ...
M and N, a
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
f \colon M \rightarrow N is called a diffeomorphism if it is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. They are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable.


Diffeomorphisms of subsets of manifolds

Given a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
X of a manifold M and a subset Y of a manifold N, a function f:X\to Y is said to be smooth if for all p in X there is a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
U\subset M of p and a smooth function g:U\to N such that the restrictions agree: g_ = f_ (note that g is an extension of f). The function f is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.


Local description

; Hadamard-Caccioppoli Theorem If U, V are
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
open subsets In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
of \R^n such that V is
simply connected 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 spac ...
, a
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
map f:U\to V is a diffeomorphism if it is
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
and if the differential Df_x:\R^n\to\R^n is bijective (and hence a
linear isomorphism In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
) at each point x in U.
; First remark It is essential for V to be
simply connected 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 spac ...
for the function f to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of the
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
square function : \begin f : \R^2 \setminus \ \to \R^2 \setminus \ \\ (x,y)\mapsto(x^2-y^2,2xy). \end Then f is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
and it satisfies : \det Df_x = 4(x^2+y^2) \neq 0. Thus, though Df_x is bijective at each point, f is not invertible because it fails to be
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
(e.g. f(1,0)=(1,0)=f(-1,0)).
; Second remark Since the differential at a point (for a differentiable function) : Df_x : T_xU \to T_V is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
, it has a well-defined inverse if and only if Df_x is a bijection. The
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
representation of Df_x is the n\times n matrix of first-order
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s whose entry in the i-th row and j-th column is \partial f_i / \partial x_j. This so-called
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
is often used for explicit computations.
; Third remark Diffeomorphisms are necessarily between manifolds of the same
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
. Imagine f going from dimension n to dimension k. If n then Df_x could never be surjective, and if n>k then Df_x could never be injective. In both cases, therefore, Df_x fails to be a bijection.
; Fourth remark If Df_x is a bijection at x then f is said to be a
local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. Formal ...
(since, by continuity, Df_y will also be bijective for all y sufficiently close to x).
; Fifth remark Given a smooth map from dimension n to dimension k, if Df (or, locally, Df_x) is surjective, f is said to be a submersion (or, locally, a "local submersion"); and if Df (or, locally, Df_x) is injective, f is said to be an
immersion Immersion may refer to: The arts * "Immersion", a 2012 story by Aliette de Bodard * ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux * Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
(or, locally, a "local immersion").
; Sixth remark A differentiable bijection is ''not'' necessarily a diffeomorphism. f(x)=x^3, for example, is not a diffeomorphism from \R to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
that is not a diffeomorphism.
; Seventh remark When f is a map between ''differentiable'' manifolds, a diffeomorphic f is a stronger condition than a homeomorphic f. For a diffeomorphism, f and its inverse need to be
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
; for a homeomorphism, f and its inverse need only be
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.
f:M\to N is called a diffeomorphism if, in coordinate charts, it satisfies the definition above. More precisely: Pick any cover of M by compatible coordinate charts and do the same for N. Let \phi and \psi be charts on, respectively, M and N, with U and V as, respectively, the images of \phi and \psi. The map \psi f\phi^:U\to V is then a diffeomorphism as in the definition above, whenever f(\phi^(U))\subseteq\psi^(V).


Examples

Since any manifold can be locally parametrised, we can consider some explicit maps from \R^2 into \R^2. * Let :: f(x,y) = \left (x^2 + y^3, x^2 - y^3 \right ). : We can calculate the Jacobian matrix: :: J_f = \begin 2x & 3y^2 \\ 2x & -3y^2 \end . : The Jacobian matrix has zero
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
if and only if xy=0. We see that f could only be a diffeomorphism away from the x-axis and the y-axis. However, f is not bijective since f(x,y)=f(-x,y), and thus it cannot be a diffeomorphism. * Let :: g(x,y) = \left (a_0 + a_x + a_y + \cdots, \ b_0 + b_x + b_y + \cdots \right ) : where the a_ and b_ are arbitrary
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, and the omitted terms are of degree at least two in ''x'' and ''y''. We can calculate the Jacobian matrix at 0: :: J_g(0,0) = \begin a_ & a_ \\ b_ & b_ \end. : We see that ''g'' is a local diffeomorphism at 0 if, and only if, :: a_b_ - a_b_ \neq 0, : i.e. the linear terms in the components of ''g'' are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
as
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s. * Let :: h(x,y) = \left (\sin(x^2 + y^2), \cos(x^2 + y^2) \right ). : We can calculate the Jacobian matrix: :: J_h = \begin 2x\cos(x^2 + y^2) & 2y\cos(x^2 + y^2) \\ -2x\sin(x^2+y^2) & -2y\sin(x^2 + y^2) \end . : The Jacobian matrix has zero determinant everywhere! In fact we see that the image of ''h'' is the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
.


Surface deformations

In
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, a stress-induced transformation is called a
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defo ...
and may be described by a diffeomorphism. A diffeomorphism f:U\to V between two
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
s U and V has a Jacobian matrix Df that is an
invertible matrix In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...
. In fact, it is required that for p in U, there is a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of p in which the Jacobian Df stays
non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...
. Suppose that in a chart of the surface, f(x,y) = (u,v). The
total differential In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by :dy = f'(x)\,dx, where f'(x) is the ...
of ''u'' is :du = \frac dx + \frac dy, and similarly for ''v''. Then the image (du, dv) = (dx, dy) Df is a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
, fixing the origin, and expressible as the action of a complex number of a particular type. When (''dx'', ''dy'') is also interpreted as that type of complex number, the action is of complex multiplication in the appropriate complex number plane. As such, there is a type of angle ( Euclidean,
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
, or
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
) that is preserved in such a multiplication. Due to ''Df'' being invertible, the type of complex number is uniform over the surface. Consequently, a surface deformation or diffeomorphism of surfaces has the conformal property of preserving (the appropriate type of) angles.


Diffeomorphism group

Let M be a differentiable manifold that is
second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
and Hausdorff. The diffeomorphism group of M is the
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 ...
of all C^r diffeomorphisms of M to itself, denoted by \text^r(M) or, when r is understood, \text(M). This is a "large" group, in the sense that—provided M is not zero-dimensional—it is not
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
.


Topology

The diffeomorphism group has two natural
topologies In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
: ''weak'' and ''strong'' . When the manifold is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
, these two topologies agree. The weak topology is always
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
. When the manifold is not compact, the strong topology captures the behavior of functions "at infinity" and is not metrizable. It is, however, still Baire. Fixing a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
on M, the weak topology is the topology induced by the family of metrics : d_K(f,g) = \sup\nolimits_ d(f(x),g(x)) + \sum\nolimits_ \sup\nolimits_ \left \, D^pf(x) - D^pg(x) \right \, as K varies over compact subsets of M. Indeed, since M is \sigma-compact, there is a sequence of compact subsets K_n whose
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
is M. Then: : d(f,g) = \sum\nolimits_n 2^\frac. The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of C^r vector fields . Over a compact subset of M, this follows by fixing a Riemannian metric on M and using the exponential map for that metric. If r is finite and the manifold is compact, the space of vector fields is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. Moreover, the transition maps from one chart of this atlas to another are smooth, making the diffeomorphism group into a
Banach manifold In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below). B ...
with smooth right translations; left translations and inversion are only continuous. If r=\infty, the space of vector fields is a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
. Moreover, the transition maps are smooth, making the diffeomorphism group into a
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. More precisely, a Fréchet manifold consists of a Haus ...
and even into a regular Fréchet Lie group. If the manifold is \sigma-compact and not compact the full diffeomorphism group is not locally contractible for any of the two topologies. One has to restrict the group by controlling the deviation from the identity near infinity to obtain a diffeomorphism group which is a manifold; see .


Lie algebra

The
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of the diffeomorphism group of M consists of all vector fields on M equipped with the
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 m ...
. Somewhat formally, this is seen by making a small change to the coordinate x at each point in space: : x^ \mapsto x^ + \varepsilon h^(x) so the infinitesimal generators are the vector fields : L_ = h^(x)\frac.


Examples

* When M=G is a
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 ...
, there is a natural inclusion of G in its own diffeomorphism group via left-translation. Let \text(G) denote the diffeomorphism group of G, then there is a splitting \text(G)\simeq G\times\text(G,e), where \text(G,e) is the
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of \text(G) that fixes the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of the group. * The diffeomorphism group of Euclidean space \R^n consists of two components, consisting of the orientation-preserving and orientation-reversing diffeomorphisms. In fact, the
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. This forms a group, because the product of two invertible matrices is again invertible, ...
is a
deformation retract In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deforma ...
of the subgroup \text(\R^n,0) of diffeomorphisms fixing the origin under the map f(x)\to f(tx)/t, t\in(0,1]. In particular, the general linear group is also a deformation retract of the full diffeomorphism group. * For a finite Set (mathematics), set of points, the diffeomorphism group is simply the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
. Similarly, if M is any manifold there is a
group extension In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overs ...
0\to\text_0(M)\to\text(M)\to\Sigma(\pi_0(M)). Here \text_0(M) is the subgroup of \text(M) that preserves all the components of M, and \Sigma(\pi_0(M)) is the permutation group of the set \pi_0(M) (the components of M). Moreover, the image of the map \text(M)\to\Sigma(\pi_0(M)) is the bijections of \pi_0(M) that preserve diffeomorphism classes.


Transitivity

For a connected manifold M, the diffeomorphism group
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
transitively on M. More generally, the diffeomorphism group acts transitively on the configuration space C_k M. If M is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space F_k M and the action on M is multiply transitive .


Extensions of diffeomorphisms

In 1926,
Tibor Radó Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I. Biography Radó was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute, studying civ ...
asked whether the harmonic extension of any homeomorphism or diffeomorphism of the unit circle to the
unit disc 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: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose di ...
yields a diffeomorphism on the open disc. An elegant proof was provided shortly afterwards by
Hellmuth Kneser Hellmuth Kneser (16 April 1898 – 23 August 1973) was a Baltic German mathematician, who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition for 3-manifold ...
. In 1945,
Gustave Choquet Gustave Choquet (; 1 March 1915 – 14 November 2006) was a French mathematician. Choquet was born in Solesmes, Nord. His contributions include work in functional analysis, potential theory, topology and measure theory. He is known for creati ...
, apparently unaware of this result, produced a completely different proof. The (orientation-preserving) diffeomorphism group of the circle is pathwise connected. This can be seen by noting that any such diffeomorphism can be lifted to a diffeomorphism f of the reals satisfying
(x+1)=f(x)+1 X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
/math>; this space is convex and hence path-connected. A smooth, eventually constant path to the identity gives a second more elementary way of extending a diffeomorphism from the circle to the open unit disc (a special case of the
Alexander trick Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander. Statement Two homeomorphisms of the ''n''-dimensional ball D^n which agree on the boundary sphere S^ are isotopic. Mo ...
). Moreover, the diffeomorphism group of the circle has the homotopy-type of the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
O(2). The corresponding extension problem for diffeomorphisms of higher-dimensional spheres S^ was much studied in the 1950s and 1960s, with notable contributions from
René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became w ...
,
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
and
Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...
. An obstruction to such extensions is given by the finite
abelian group In mathematics, 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. That is, the group operation is commut ...
\Gamma_n, the " group of twisted spheres", defined as the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the abelian
component group In mathematics, specifically group theory, the identity component of a group (mathematics) , group ''G'' refers to several closely related notions of the largest connected space , connected subgroup of ''G'' containing the identity element. In p ...
of the diffeomorphism group by the subgroup of classes extending to diffeomorphisms of the ball B^n.


Connectedness

For manifolds, the diffeomorphism group is usually not connected. Its component group is called the
mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mot ...
. In dimension 2 (i.e.
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
s), the mapping class group is a
finitely presented group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
generated by
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). Definition Suppose that ''c'' is a simple closed curve in a closed, orientable surface ''S''. Let ' ...
s ( Dehn,
Lickorish William Bernard Raymond Lickorish (born 19 February 1938) is a mathematician. He is emeritus professor of geometric topology in the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, and also an emeritus fellow ...
,
Hatcher Hatcher is a surname. Notable people with the surname include: *Allen Hatcher (born 1944), U.S. mathematician * Anna Granville Hatcher (1905–1978), U.S. linguist *Edwin Starr (born Charles Edwin Hatcher, 1942–2003), U.S. soul singer * Chris Hat ...
).
Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...
and
Jakob Nielsen Jacob or Jakob Nielsen may refer to: * Jacob Nielsen, Count of Halland (died c. 1309), great grandson of Valdemar II of Denmark * , Norway (1768-1822) * Jakob Nielsen (mathematician) (1890–1959), Danish mathematician known for work on automorphi ...
showed that it can be identified with the
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. The outer automorphism group is usually denoted . If is trivial and has a t ...
of the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the surface.
William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston ...
refined this analysis by classifying elements of the mapping class group into three types: those equivalent to a periodic diffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curve invariant; and those equivalent to pseudo-Anosov diffeomorphisms. In the case of the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
S^1\times S^1=\R^2/\Z^2, the mapping class group is simply the
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
\text(2,\Z) and the classification becomes classical in terms of
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
, parabolic and
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
matrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on a
compactification Compactification may refer to: * Compactification (mathematics), making a topological space compact * Compactification (physics), the "curling up" of extra dimensions in string theory See also * Compaction (disambiguation) Compaction may refer t ...
of
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. Teichmülle ...
; as this enlarged space was homeomorphic to a closed ball, the
Brouwer fixed-point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest ...
became applicable. Smale
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
d that if M is an
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
smooth closed manifold, the
identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity componen ...
of the group of orientation-preserving diffeomorphisms is
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
. This had first been proved for a product of circles by Michel Herman; it was proved in full generality by Thurston.


Homotopy types

* The diffeomorphism group of S^2 has the homotopy-type of the subgroup O(3). This was proven by Steve Smale. * The diffeomorphism group of the torus has the homotopy-type of its linear
automorphism 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 ...
s: S^1\times S^1\times\text(2,\Z). * The diffeomorphism groups of orientable surfaces of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
g>1 have the homotopy-type of their mapping class groups (i.e. the components are contractible). * The homotopy-type of the diffeomorphism groups of 3-manifolds are fairly well understood via the work of Ivanov, Hatcher, Gabai and Rubinstein, although there are a few outstanding open cases (primarily 3-manifolds with finite
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
s). * The homotopy-type of diffeomorphism groups of n-manifolds for n>3 are poorly understood. For example, it is an open problem whether or not \text(S^4) has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided n>6, \text(S^n) does not have the homotopy-type of a finite
CW-complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
.


Homeomorphism and diffeomorphism

Since every diffeomorphism is a homeomorphism, every diffeomorphic manifolds are homeomorphic, but the converse is not true. While it is easy to find homeomorphisms that are non-diffeomorphisms, it is more difficult to find a pair of
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
manifolds that are not diffeomorphic. In dimensions 1, 2 and 3, any pair of homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairs have been found. The first such example was constructed by
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
in dimension 7. He constructed a smooth 7-dimensional manifold (called now
Milnor's sphere In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnorpg 14 was trying to understand the structure of (n-1)-connected manifolds of dimension 2n (since n-connected 2n-manifolds are homeomorphic to sph ...
) that is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are, in fact, 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is the total space of a
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
over the 4-sphere with the
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
as the fiber). More unusual phenomena occur for
4-manifold In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. T ...
s. In the early 1980s, a combination of results due to
Simon Donaldson Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. He i ...
and
Michael Freedman Michael Hartley Freedman (born April 21, 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional gen ...
led to the discovery of
exotic Exotic may refer to: Mathematics and physics * Exotic R4, a differentiable 4-manifold, homeomorphic but not diffeomorphic to the Euclidean space R4 * Exotic sphere, a differentiable ''n''-manifold, homeomorphic but not diffeomorphic to the ordina ...
\R^4: there are uncountably many pairwise non-diffeomorphic open subsets of \R^4 each of which is homeomorphic to \R^4, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to \R^4 that do not embed smoothly in \R^4.


See also

*
Anosov diffeomorphism In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...
such as Arnold's cat map *
Diffeo anomaly In theoretical physics, a gravitational anomaly is an example of a gauge anomaly: it is an effect of quantum mechanics — usually a one-loop diagram—that invalidates the general covariance of a theory of general relativity combined with som ...
also known as a
gravitational anomaly In theoretical physics, a gravitational anomaly is an example of a gauge anomaly: it is an effect of quantum mechanics — usually a one-loop diagram—that invalidates the general covariance of a theory of general relativity combined with som ...
, a type anomaly in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
*
Diffeology In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are. The concept was first introduced by Jean-Marie Souriau in the 1980s under ...
, smooth parameterizations on a set, which makes a diffeological space * Diffeomorphometry, metric study of shape and form in computational anatomy *
Étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy th ...
*
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. For ...
*
Local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. Formal ...
* Superdiffeomorphism


Notes


References

* * * * * * * * * * * * {{Manifolds Mathematical physics