Inmathematics, adiffeomorphism is anisomorphism ofdifferentiable manifolds. It is aninvertible function that maps one differentiable manifold to another such that both the function and its inverse arecontinuously differentiable.

Given two differentiable manifolds${\displaystyle M}$ and${\displaystyle N}$, adifferentiable map${\displaystyle f:M\to N}$ is adiffeomorphism if it is abijection and its inverse${\displaystyle f^{-1}:N\to M}$ is differentiable as well. If these functions are${\displaystyle r}$ times continuously differentiable,${\displaystyle f}$ is called a${\displaystyle C^r}$-diffeomorphism.

Two manifolds${\displaystyle M}$ and${\displaystyle N}$ arediffeomorphic (usually denoted${\displaystyle M\simeq N}$) if there is a diffeomorphism${\displaystyle f}$ from${\displaystyle M}$ to${\displaystyle N}$. Two${\displaystyle C^r}$-differentiable manifolds are${\displaystyle C^r}$-diffeomorphic if there is an${\displaystyle r}$ times continuously differentiable bijective map between them whose inverse is also${\displaystyle r}$ times continuously differentiable.

Given asubset${\displaystyle X}$ of a manifold${\displaystyle M}$ and a subset${\displaystyle Y}$ of a manifold${\displaystyle N}$, a function${\displaystyle f:X\to Y}$ is said to be smooth if for all${\displaystyle p}$ in${\displaystyle X}$ there is aneighborhood${\displaystyle U\subset M}$ of${\displaystyle p}$ and a smooth function${\displaystyle g:U\to N}$ such that therestrictions agree:${\displaystyle g_{|U\cap X}=f_{|U\cap X}}$ (note that${\displaystyle g}$ is an extension of${\displaystyle f}$). The function${\displaystyle f}$ is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.

Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem:^[1]

If${\displaystyle U}$,${\displaystyle V}$ areconnectedopen subsets of${\displaystyle {\mathbb{R}}^n}$ such that${\displaystyle V}$ issimply connected, a differentiable map${\displaystyle f:U\to V}$ is a diffeomorphism if it isproper and if thedifferential${\displaystyle D{f}_x:{\mathbb{R}}^n\to {\mathbb{R}}^n}$ is bijective (and hence alinear isomorphism) at each point${\displaystyle x}$ in${\displaystyle U}$.

Some remarks:

It is essential for${\displaystyle V}$ to besimply connected for the function${\displaystyle f}$ to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of thecomplex square function

${\displaystyle \{\begin{array}{l}f:{\mathbb{R}}^2\setminus \{(0,0)\}\to {\mathbb{R}}^2\setminus \{(0,0)\}\\ (x,y)\mapsto (x^2-y^2,2xy).\end{array}}$

Then${\displaystyle f}$ issurjective and it satisfies

${\displaystyle detD{f}_x=4(x^2+y^2)\ne 0.}$

Thus, though${\displaystyle D{f}_x}$ is bijective at each point,${\displaystyle f}$ is not invertible because it fails to beinjective (e.g.${\displaystyle f(1,0)=(1,0)=f(-1,0)}$).

Since the differential at a point (for a differentiable function)

${\displaystyle D{f}_x:T_{x}U\to T_{f(x)}V}$

is alinear map, it has a well-defined inverse if and only if${\displaystyle D{f}_x}$ is a bijection. Thematrix representation of${\displaystyle D{f}_x}$ is the${\displaystyle n\times n}$ matrix of first-orderpartial derivatives whose entry in the${\displaystyle i}$-th row and${\displaystyle j}$-th column is${\displaystyle \mathrm{\partial }{f}_i/\mathrm{\partial }{x}_j}$. This so-calledJacobian matrix is often used for explicit computations.

Diffeomorphisms are necessarily between manifolds of the samedimension. Imagine${\displaystyle f}$ going from dimension${\displaystyle n}$ to dimension${\displaystyle k}$. If then${\displaystyle D{f}_x}$ could never be surjective, and if${\displaystyle n>k}$ then${\displaystyle D{f}_x}$ could never be injective. In both cases, therefore,${\displaystyle D{f}_x}$ fails to be a bijection.

If${\displaystyle D{f}_x}$ is a bijection at${\displaystyle x}$ then${\displaystyle f}$ is said to be alocal diffeomorphism (since, by continuity,${\displaystyle D{f}_y}$ will also be bijective for all${\displaystyle y}$ sufficiently close to${\displaystyle x}$).

Given a smooth map from dimension${\displaystyle n}$ to dimension${\displaystyle k}$, if${\displaystyle Df}$ (or, locally,${\displaystyle D{f}_x}$) is surjective,${\displaystyle f}$ is said to be asubmersion (or, locally, a "local submersion"); and if${\displaystyle Df}$ (or, locally,${\displaystyle D{f}_x}$) is injective,${\displaystyle f}$ is said to be animmersion (or, locally, a "local immersion").

A differentiable bijection isnot necessarily a diffeomorphism.${\displaystyle f(x)=x^3}$, for example, is not a diffeomorphism from${\displaystyle \mathbb{R}}$ to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of ahomeomorphism that is not a diffeomorphism.

When${\displaystyle f}$ is a map between differentiable manifolds, a diffeomorphic${\displaystyle f}$ is a stronger condition than a homeomorphic${\displaystyle f}$. For a diffeomorphism,${\displaystyle f}$ and its inverse need to bedifferentiable; for a homeomorphism,${\displaystyle f}$ and its inverse need only becontinuous. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.

${\displaystyle f:M\to N}$ is a diffeomorphism if, incoordinate charts, it satisfies the definition above. More precisely: Pick any cover of${\displaystyle M}$ by compatiblecoordinate charts and do the same for${\displaystyle N}$. Let${\displaystyle \varphi }$ and${\displaystyle \psi }$ be charts on, respectively,${\displaystyle M}$ and${\displaystyle N}$, with${\displaystyle U}$ and${\displaystyle V}$ as, respectively, the images of${\displaystyle \varphi }$ and${\displaystyle \psi }$. The map${\displaystyle \psi f{\varphi }^{-1}:U\to V}$ is then a diffeomorphism as in the definition above, whenever${\displaystyle f({\varphi }^{-1}(U))\subseteq {\psi }^{-1}(V)}$.

Since any manifold can be locally parametrised, we can consider some explicit maps from${\displaystyle {\mathbb{R}}^2}$ into${\displaystyle {\mathbb{R}}^2}$.

• Let

${\displaystyle f(x,y)=\left(x^2+y^3,x^2-y^3\right).}$

We can calculate the Jacobian matrix:

The Jacobian matrix has zerodeterminant if and only if${\displaystyle xy=0}$. We see that${\displaystyle f}$ could only be a diffeomorphism away from the${\displaystyle x}$-axis and the${\displaystyle y}$-axis. However,${\displaystyle f}$ is not bijective since${\displaystyle f(x,y)=f(-x,y)}$, and thus it cannot be a diffeomorphism.

• Let

${\displaystyle g(x,y)=\left(a_0+a_{1,0}x+a_{0,1}y+\cdots ,b_0+b_{1,0}x+b_{0,1}y+\cdots \right)}$

where the${\displaystyle a_{i,j}}$ and${\displaystyle b_{i,j}}$ are arbitraryreal numbers, and the omitted terms are of degree at least two inx andy. We can calculate the Jacobian matrix at0:

We see thatg is a local diffeomorphism at0 if, and only if,

${\displaystyle a_{1,0}{b}_{0,1}-a_{0,1}{b}_{1,0}\ne 0,}$

i.e. the linear terms in the components ofg arelinearly independent aspolynomials.

• Let

${\displaystyle h(x,y)=\left(\sin (x^2+y^2),\cos (x^2+y^2)\right).}$

We can calculate the Jacobian matrix:

The Jacobian matrix has zero determinant everywhere! In fact we see that the image ofh is theunit circle.

Inmechanics, a stress-induced transformation is called adeformation and may be described by a diffeomorphism.A diffeomorphism${\displaystyle f:U\to V}$ between twosurfaces${\displaystyle U}$ and${\displaystyle V}$ has a Jacobian matrix${\displaystyle Df}$ that is aninvertible matrix. In fact, it is required that for${\displaystyle p}$ in${\displaystyle U}$, there is aneighborhood of${\displaystyle p}$ in which the Jacobian${\displaystyle Df}$ staysnon-singular. Suppose that in a chart of the surface,${\displaystyle f(x,y)=(u,v).}$

Thetotal differential ofu is

${\displaystyle du=\frac{\mathrm{\partial }u}{\mathrm{\partial }x}dx+\frac{\mathrm{\partial }u}{\mathrm{\partial }y}dy}$, and similarly forv.

Then the image${\displaystyle (du,dv)=(dx,dy)Df}$ is alinear transformation, 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, orslope) that is preserved in such a multiplication. Due toDf being invertible, the type of complex number is uniform over the surface. Consequently, a surface deformation or diffeomorphism of surfaces has theconformal property of preserving (the appropriate type of) angles.

Let${\displaystyle M}$ be a differentiable manifold that issecond-countable andHausdorff. Thediffeomorphism group of${\displaystyle M}$ is thegroup of all${\displaystyle C^r}$ diffeomorphisms of${\displaystyle M}$ to itself, denoted by${\displaystyle {\text{Diff}}^r(M)}$ or, when${\displaystyle r}$ is understood,${\displaystyle \text{Diff}(M)}$. This is a "large" group, in the sense that—provided${\displaystyle M}$ is not zero-dimensional—it is notlocally compact.

The diffeomorphism group has two naturaltopologies:weak andstrong (Hirsch 1997). When the manifold iscompact, these two topologies agree. The weak topology is alwaysmetrizable. When the manifold is not compact, the strong topology captures the behavior of functions "at infinity" and is not metrizable. It is, however, stillBaire.

Fixing aRiemannian metric on${\displaystyle M}$, the weak topology is the topology induced by the family of metrics

${\displaystyle d_K(f,g)={sup}_{x\in K}d(f(x),g(x))+{\sum }_{1\le p\le r}{sup}_{x\in K}\Vert D^{p}f(x)-D^{p}g(x)\Vert }$

as${\displaystyle K}$ varies over compact subsets of${\displaystyle M}$. Indeed, since${\displaystyle M}$ is${\displaystyle \sigma }$-compact, there is a sequence of compact subsets${\displaystyle K_n}$ whoseunion is${\displaystyle M}$. Then:

${\displaystyle d(f,g)={\sum }_n{2}^{-n}\frac{d_{K_n}(f,g)}{1+d_{K_n}(f,g)}.}$

The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of${\displaystyle C^r}$ vector fields (Leslie 1967). Over a compact subset of${\displaystyle M}$, this follows by fixing a Riemannian metric on${\displaystyle M}$ and using theexponential map for that metric. If${\displaystyle r}$ is finite and the manifold is compact, the space of vector fields is aBanach space. Moreover, the transition maps from one chart of this atlas to another are smooth, making the diffeomorphism group into aBanach manifold with smooth right translations; left translations and inversion are only continuous. If${\displaystyle r=\mathrm{\infty }}$, the space of vector fields is aFréchet space. Moreover, the transition maps are smooth, making the diffeomorphism group into aFréchet manifold and even into aregular Fréchet Lie group. If the manifold is${\displaystyle \sigma }$-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 (Michor & Mumford 2013).

TheLie algebra of the diffeomorphism group of${\displaystyle M}$ consists of allvector fields on${\displaystyle M}$ equipped with theLie bracket of vector fields. Somewhat formally, this is seen by making a small change to the coordinate${\displaystyle x}$ at each point in space:

${\displaystyle x^{\mu }\mapsto x^{\mu }+\epsilon h^{\mu }(x)}$

so the infinitesimal generators are the vector fields

${\displaystyle L_h=h^{\mu }(x)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mu }}.}$

• When${\displaystyle M=G}$ is aLie group, there is a natural inclusion of${\displaystyle G}$ in its own diffeomorphism group via left-translation. Let${\displaystyle \text{Diff}(G)}$ denote the diffeomorphism group of${\displaystyle G}$, then there is a splitting${\displaystyle \text{Diff}(G)\simeq G\times \text{Diff}(G,e)}$, where${\displaystyle \text{Diff}(G,e)}$ is thesubgroup of${\displaystyle \text{Diff}(G)}$ that fixes theidentity element of the group.
• The diffeomorphism group of Euclidean space${\displaystyle {\mathbb{R}}^n}$ consists of two components, consisting of the orientation-preserving and orientation-reversing diffeomorphisms. In fact, thegeneral linear group is adeformation retract of the subgroup${\displaystyle \text{Diff}({\mathbb{R}}^n,0)}$ of diffeomorphisms fixing the origin under the map${\displaystyle f(x)\to f(tx)/t,t\in (0,1]}$. In particular, the general linear group is also a deformation retract of the full diffeomorphism group.
• For a finiteset of points, the diffeomorphism group is simply thesymmetric group. Similarly, if${\displaystyle M}$ is any manifold there is agroup extension${\displaystyle 0\to {\text{Diff}}_0(M)\to \text{Diff}(M)\to \mathrm{\Sigma }({\pi }_0(M))}$. Here${\displaystyle {\text{Diff}}_0(M)}$ is the subgroup of${\displaystyle \text{Diff}(M)}$ that preserves all the components of${\displaystyle M}$, and${\displaystyle \mathrm{\Sigma }({\pi }_0(M))}$ is the permutation group of the set${\displaystyle {\pi }_0(M)}$ (the components of${\displaystyle M}$). Moreover, the image of the map${\displaystyle \text{Diff}(M)\to \mathrm{\Sigma }({\pi }_0(M))}$ is the bijections of${\displaystyle {\pi }_0(M)}$ that preserve diffeomorphism classes.

For a connected manifold${\displaystyle M}$, the diffeomorphism groupactstransitively on${\displaystyle M}$. More generally, the diffeomorphism group acts transitively on theconfiguration space${\displaystyle C_{k}M}$. If${\displaystyle M}$ is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space${\displaystyle F_{k}M}$ and the action on${\displaystyle M}$ ismultiply transitive (Banyaga 1997, p. 29).

In 1926,Tibor Radó asked whether theharmonic extension of any homeomorphism or diffeomorphism of the unit circle to theunit disc yields a diffeomorphism on the open disc. An elegant proof was provided shortly afterwards byHellmuth Kneser. In 1945,Gustave Choquet, 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${\displaystyle f}$ of the reals satisfying${\displaystyle [f(x+1)=f(x)+1]}$; 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 theAlexander trick). Moreover, the diffeomorphism group of the circle has the homotopy-type of theorthogonal group${\displaystyle O(2)}$.

The corresponding extension problem for diffeomorphisms of higher-dimensional spheres${\displaystyle S^{n-1}}$ was much studied in the 1950s and 1960s, with notable contributions fromRené Thom,John Milnor andStephen Smale. An obstruction to such extensions is given by the finiteabelian group${\displaystyle {\mathrm{\Gamma }}_n}$, the "group of twisted spheres", defined as thequotient of the abeliancomponent group of the diffeomorphism group by the subgroup of classes extending to diffeomorphisms of the ball${\displaystyle B^n}$.

For manifolds, the diffeomorphism group is usually not connected. Its component group is called themapping class group. In dimension 2 (i.e.surfaces), the mapping class group is afinitely presented group generated byDehn twists; this has been proved byMax Dehn,W. B. R. Lickorish, andAllen Hatcher).^{[citation needed]} Max Dehn andJakob Nielsen showed that it can be identified with theouter automorphism group of thefundamental group of the surface.

William Thurston refined this analysis byclassifying elements of the mapping class group into three types: those equivalent to aperiodic diffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curve invariant; and those equivalent topseudo-Anosov diffeomorphisms. In the case of thetorus${\displaystyle S^1\times S^1={\mathbb{R}}^2/{\mathbb{Z}}^2}$, the mapping class group is simply themodular group${\displaystyle \text{SL}(2,\mathbb{Z})}$ and the classification becomes classical in terms ofelliptic,parabolic andhyperbolic matrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on acompactification ofTeichmüller space; as this enlarged space was homeomorphic to a closed ball, theBrouwer fixed-point theorem became applicable. Smaleconjectured that if${\displaystyle M}$ is anoriented smooth closed manifold, theidentity component of the group of orientation-preserving diffeomorphisms issimple. This had first been proved for a product of circles byMichel Herman; it was proved in full generality by Thurston.

• The diffeomorphism group of${\displaystyle S^2}$ has the homotopy-type of the subgroup${\displaystyle O(3)}$. This was proven by Steve Smale.^[2]
• The diffeomorphism group of the torus has the homotopy-type of its linearautomorphisms:${\displaystyle S^1\times S^1\times \text{GL}(2,\mathbb{Z})}$.
• The diffeomorphism groups of orientable surfaces ofgenus${\displaystyle g>1}$ 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 finitefundamental groups).
• The homotopy-type of diffeomorphism groups of${\displaystyle n}$-manifolds for${\displaystyle n>3}$ are poorly understood. For example, it is an open problem whether or not${\displaystyle \text{Diff}(S^4)}$ has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided${\displaystyle n>6}$,${\displaystyle \text{Diff}(S^n)}$ does not have the homotopy-type of a finiteCW-complex.

Since every diffeomorphism is a homeomorphism, given a pair of manifolds which are diffeomorphic to each other they are in particularhomeomorphic to each other. The converse is not true in general.

While it is easy to find homeomorphisms that are not diffeomorphisms, it is more difficult to find a pair of homeomorphic 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 exist. The first such example was constructed byJohn Milnor in dimension 7. He constructed a smooth 7-dimensional manifold (called nowMilnor's sphere) 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 afiber bundle over the 4-sphere with the3-sphere as the fiber).

More unusual phenomena occur for4-manifolds. In the early 1980s, a combination of results due toSimon Donaldson andMichael Freedman led to the discovery ofexotic${\displaystyle {\mathbb{R}}^4}$: there areuncountably many pairwise non-diffeomorphic open subsets of${\displaystyle {\mathbb{R}}^4}$ each of which is homeomorphic to${\displaystyle {\mathbb{R}}^4}$, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to${\displaystyle {\mathbb{R}}^4}$ that do notembed smoothly in${\displaystyle {\mathbb{R}}^4}$.

• Anosov diffeomorphism such asArnold's cat map
• Diffeo anomaly also known as agravitational anomaly, a typeanomaly inquantum mechanics
• Diffeology, smooth parameterizations on a set, which makes a diffeological space
• Diffeomorphometry, metric study of shape and form in computational anatomy
• Étale morphism
• Large diffeomorphism
• Local diffeomorphism
• Superdiffeomorphism

• Krantz, Steven G.; Parks, Harold R. (2013).The implicit function theorem: history, theory, and applications. Modern Birkhäuser classics. Boston.ISBN 978-1-4614-5980-4.{{cite book}}: CS1 maint: location missing publisher (link)
• Chaudhuri, Shyamoli; Kawai, Hikaru; Tye, S.-H. Henry (1987-08-15)."Path-integral formulation of closed strings"(PDF).Physical Review D.36 (4):1148–1168.Bibcode:1987PhRvD..36.1148C.doi:10.1103/physrevd.36.1148.ISSN 0556-2821.PMID 9958280.S2CID 41709882.Archived(PDF) from the original on 2018-07-21.
• Banyaga, Augustin (1997),The structure of classical diffeomorphism groups, Mathematics and its Applications, vol. 400, Kluwer Academic,ISBN 0-7923-4475-8
• Duren, Peter L. (2004),Harmonic Mappings in the Plane, Cambridge Mathematical Tracts, vol. 156, Cambridge University Press,ISBN 0-521-64121-7
• "Diffeomorphism",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Hirsch, Morris (1997),Differential Topology, Berlin, New York:Springer-Verlag,ISBN 978-0-387-90148-0
• Kriegl, Andreas; Michor, Peter (1997),The convenient setting of global analysis, Mathematical Surveys and Monographs, vol. 53, American Mathematical Society,ISBN 0-8218-0780-3
• Leslie, J. A. (1967), "On a differential structure for the group of diffeomorphisms",Topology,6 (2):263–271,doi:10.1016/0040-9383(67)90038-9,ISSN 0040-9383,MR 0210147
• Michor, Peter W.; Mumford, David (2013), "A zoo of diffeomorphism groups onRⁿ.",Annals of Global Analysis and Geometry,44 (4):529–540,arXiv:1211.5704,doi:10.1007/s10455-013-9380-2,S2CID 118624866
• Milnor, John W. (2007),Collected Works Vol. III, Differential Topology, American Mathematical Society,ISBN 978-0-8218-4230-0
• Omori, Hideki (1997),Infinite-dimensional Lie groups, Translations of Mathematical Monographs, vol. 158, American Mathematical Society,ISBN 0-8218-4575-6
• Kneser, Hellmuth (1926), "Lösung der Aufgabe 41.",Jahresbericht der Deutschen Mathematiker-Vereinigung (in German),35 (2): 123

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