In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definition Given two differentiable manifolds M {\displaystyle M} and N {\displaystyle N} , a continuously differentiable map f : M → N {\displaystyle f\colon M\rightarrow N} is a diffeomorphism if it is a bijection and its inverse f − 1 : N → M {\displaystyle f^{-1}\colon N\rightarrow M} is differentiable as well. If these functions are r {\displaystyle r} times continuously differentiable, f {\displaystyle f} is called a C r {\displaystyle C^{r}} -diffeomorphism. Two manifolds M {\displaystyle M} and N {\displaystyle N} are diffeomorphic (usually denoted M ≃ N {\displaystyle M\simeq N} ) if there is a diffeomorphism f {\displaystyle f} from M {\displaystyle M} to N {\displaystyle N} . Two C r {\displaystyle C^{r}} -differentiable manifolds are C r {\displaystyle C^{r}} -diffeomorphic if there is an r {\displaystyle r} times continuously differentiable bijective map between them whose inverse is also r {\displaystyle r} times continuously differentiable. A C 1 {\displaystyle C^{1}} -diffeomorphism is simply a diffeomorphism, and a C 0 {\displaystyle C^{0}} -diffeomorphism is a homeomorphism.
Diffeomorphisms of subsets of manifolds Given a subset X {\displaystyle X} of a manifold M {\displaystyle M} and a subset Y {\displaystyle Y} of a manifold N {\displaystyle N} , a function f : X → Y {\displaystyle f:X\to Y} is said to be smooth if for all p {\displaystyle p} in X {\displaystyle X} there is a neighborhood U ⊂ M {\displaystyle U\subset M} of p {\displaystyle p} and a smooth function g : U → N {\displaystyle g:U\to N} such that the restrictions agree: g | U ∩ X = f | U ∩ X {\displaystyle g_{|U\cap X}=f_{|U\cap X}} (note that g {\displaystyle g} is an extension of f {\displaystyle f} ). The function f {\displaystyle f} is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.
Local description Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem: If U {\displaystyle U} , V {\displaystyle V} are connected open subsets of R n {\displaystyle \mathbb {R} ^{n}} such that V {\displaystyle V} is simply connected, a differentiable map f : U → V {\displaystyle f:U\to V} is a diffeomorphism if it is proper and if the differential D f x : R n → R n {\displaystyle Df_{x}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} is bijective (and hence a linear isomorphism) at each point x {\displaystyle x} in U {\displaystyle U} . Some remarks: It is essential for V {\displaystyle V} to be simply connected for the function f {\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 the complex square function
{ f : R 2 ∖ { ( 0 , 0 ) } → R 2 ∖ { ( 0 , 0 ) } ( x , y ) ↦ ( x 2 − y 2 , 2 x y ) . {\displaystyle {\begin{cases}f:\mathbb {R} ^{2}\setminus \{(0,0)\}\to \mathbb {R} ^{2}\setminus \{(0,0)\}\\(x,y)\mapsto (x^{2}-y^{2},2xy).\end{cases}}}
Then f {\displaystyle f} is surjective and it satisfies
det D f x = 4 ( x 2 + y 2 ) ≠ 0. {\displaystyle \det Df_{x}=4(x^{2}+y^{2})\neq 0.}
Thus, though D f x {\displaystyle Df_{x}} is bijective at each point, f {\displaystyle f} is not invertible because it fails to be injective (e.g. f ( 1 , 0 ) = ( 1 , 0 ) = f ( − 1 , 0 ) {\displaystyle f(1,0)=(1,0)=f(-1,0)} ). Since the differential at a point (for a differentiable function)
D f x : T x U → T f ( x ) V {\displaystyle Df_{x}:T_{x}U\to T_{f(x)}V}
is a linear map, it has a well-defined inverse if and only if D f x {\displaystyle Df_{x}} is a bijection. The matrix representation of D f x {\displaystyle Df_{x}} is the n × n {\displaystyle n\times n} matrix of first-order partial derivatives whose entry in the i {\displaystyle i} -th row and j {\displaystyle j} -th column is ∂ f i / ∂ x j {\displaystyle \partial f_{i}/\partial x_{j}} . This so-called Jacobian matrix is often used for explicit computations. Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine f {\displaystyle f} going from dimension n {\displaystyle n} to dimension k {\displaystyle k} . If n < k {\displaystyle n<k} then D f x {\displaystyle Df_{x}} could never be surjective, and if n > k {\displaystyle n>k} then D f x {\displaystyle Df_{x}} could never be injective. In both cases, therefore, D f x {\displaystyle Df_{x}} fails to be a bijection. If D f x {\displaystyle Df_{x}} is a bijection at x {\displaystyle x} then f {\displaystyle f} is said to be a local diffeomorphism (since, by continuity, D f y {\displaystyle Df_{y}} will also be bijective for all y {\displaystyle y} sufficiently close to x {\displaystyle x} ). Given a smooth map from dimension n {\displaystyle n} to dimension k {\displaystyle k} , if D f {\displaystyle Df} (or, locally, D f x {\displaystyle Df_{x}} ) is surjective, f {\displaystyle f} is said to be a submersion (or, locally, a "local submersion"); and if D f {\displaystyle Df} (or, locally, D f x {\displaystyle Df_{x}} ) is injective, f {\displaystyle f} is said to be an immersion (or, locally, a "local immersion"). A differentiable bijection is not necessarily a diffeomorphism. f ( x ) = x 3 {\displaystyle f(x)=x^{3}} , for example, is not a diffeomorphism from R {\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 a homeomorphism that is not a diffeomorphism. When f {\displaystyle f} is a map between differentiable manifolds, a diffeomorphic f {\displaystyle f} is a stronger condition than a homeomorphic f {\displaystyle f} . For a diffeomorphism, f {\displaystyle f} and its inverse need to be differentiable; for a homeomorphism, f {\displaystyle f} and its inverse need only be continuous. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.
f : M → N {\displaystyle f:M\to N} is a diffeomorphism if, in coordinate charts, it satisfies the definition above. More precisely: Pick any cover of M {\displaystyle M} by compatible coordinate charts and do the same for N {\displaystyle N} . Let ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } be charts on, respectively, M {\displaystyle M} and N {\displaystyle N} , with U {\displaystyle U} and V {\displaystyle V} as, respectively, the images of ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } . The map ψ f ϕ − 1 : U → V {\displaystyle \psi f\phi ^{-1}:U\to V} is then a diffeomorphism as in the definition above, whenever f ( ϕ − 1 ( U ) ) ⊆ ψ − 1 ( V ) {\displaystyle f(\phi ^{-1}(U))\subseteq \psi ^{-1}(V)} .
Examples Since any manifold can be locally parametrised, we can consider some explicit maps from R 2 {\displaystyle \mathbb {R} ^{2}} into R 2 {\displaystyle \mathbb {R} ^{2}} .
Let
f ( x , y ) = ( x 2 + y 3 , x 2 − y 3 ) . {\displaystyle f(x,y)=\left(x^{2}+y^{3},x^{2}-y^{3}\right).}
We can calculate the Jacobian matrix:
J f = ( 2 x 3 y 2 2 x − 3 y 2 ) . {\displaystyle J_{f}={\begin{pmatrix}2x&3y^{2}\\2x&-3y^{2}\end{pmatrix}}.}
The Jacobian matrix has zero determinant if and only if x y = 0 {\displaystyle xy=0} . We see that f {\displaystyle f} could only be a diffeomorphism away from the x {\displaystyle x} -axis and the y {\displaystyle y} -axis. However, f {\displaystyle f} is not bijective since f ( x , y ) = f ( − x , y ) {\displaystyle f(x,y)=f(-x,y)} , and thus it cannot be a diffeomorphism. Let
g ( x , y ) = ( a 0 + a 1 , 0 x + a 0 , 1 y + ⋯ , b 0 + b 1 , 0 x + b 0 , 1 y + ⋯ ) {\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 a i , j {\displaystyle a_{i,j}} and b i , j {\displaystyle b_{i,j}} are arbitrary real numbers, 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 ) = ( a 1 , 0 a 0 , 1 b 1 , 0 b 0 , 1 ) . {\displaystyle J_{g}(0,0)={\begin{pmatrix}a_{1,0}&a_{0,1}\\b_{1,0}&b_{0,1}\end{pmatrix}}.}
We see that g is a local diffeomorphism at 0 if, and only if,
a 1 , 0 b 0 , 1 − a 0 , 1 b 1 , 0 ≠ 0 , {\displaystyle a_{1,0}b_{0,1}-a_{0,1}b_{1,0}\neq 0,}
i.e. the linear terms in the components of g are linearly independent as polynomials. Let
h ( x , y ) = ( sin ( x 2 + y 2 ) , cos ( x 2 + y 2 ) ) . {\displaystyle h(x,y)=\left(\sin(x^{2}+y^{2}),\cos(x^{2}+y^{2})\right).}
We can calculate the Jacobian matrix:
J h = ( 2 x cos ( x 2 + y 2 ) 2 y cos ( x 2 + y 2 ) − 2 x sin ( x 2 + y 2 ) − 2 y sin ( x 2 + y 2 ) ) . {\displaystyle J_{h}={\begin{pmatrix}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{pmatrix}}.}
The Jacobian matrix has zero determinant everywhere! In fact we see that the image of h is the unit circle.
Surface deformations In mechanics, a stress-induced transformation is called a deformation and may be described by a diffeomorphism. A diffeomorphism f : U → V {\displaystyle f:U\to V} between two surfaces U {\displaystyle U} and V {\displaystyle V} has a Jacobian matrix D f {\displaystyle Df} that is an invertible matrix. In fact, it is required that for p {\displaystyle p} in U {\displaystyle U} , there is a neighborhood of p {\displaystyle p} in which the Jacobian D f {\displaystyle Df} stays non-singular. Suppose that in a chart of the surface, f ( x , y ) = ( u , v ) . {\displaystyle f(x,y)=(u,v).}
The total differential of u is
d u = ∂ u ∂ x d x + ∂ u ∂ y d y {\displaystyle du={\frac {\partial u}{\partial x}}dx+{\frac {\partial u}{\partial y}}dy} , and similarly for v. Then the image ( d u , d v ) = ( d x , d y ) D f {\displaystyle (du,dv)=(dx,dy)Df} is a linear 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, or slope) 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 {\displaystyle M} be a differentiable manifold that is second-countable and Hausdorff. The diffeomorphism group of M {\displaystyle M} is the group of all C r {\displaystyle C^{r}} diffeomorphisms of M {\displaystyle M} to itself, denoted by Diff r ( M ) {\displaystyle {\text{Diff}}^{r}(M)} or, when r {\displaystyle r} is understood, Diff ( M ) {\displaystyle {\text{Diff}}(M)} . This is a "large" group, in the sense that—provided M {\displaystyle M} is not zero-dimensional—it is not locally compact.
Topology The diffeomorphism group has two natural topologies: weak and strong (Hirsch 1997). When the manifold is compact, these two topologies agree. The weak topology is always metrizable. 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 on M {\displaystyle M} , the weak topology is the topology induced by the family of metrics
d K ( f , g ) = sup x ∈ K d ( f ( x ) , g ( x ) ) + ∑ 1 ≤ p ≤ r sup x ∈ K ‖ D p f ( x ) − D p g ( x ) ‖ {\displaystyle d_{K}(f,g)=\sup \nolimits _{x\in K}d(f(x),g(x))+\sum \nolimits _{1\leq p\leq r}\sup \nolimits _{x\in K}\left\|D^{p}f(x)-D^{p}g(x)\right\|}
as K {\displaystyle K} varies over compact subsets of M {\displaystyle M} . Indeed, since M {\displaystyle M} is σ {\displaystyle \sigma } -compact, there is a sequence of compact subsets K n {\displaystyle K_{n}} whose union is M {\displaystyle M} . Then:
d ( f , g ) = ∑ n 2 − n d K n ( f , g ) 1 + d K n ( f , g ) . {\displaystyle d(f,g)=\sum \nolimits _{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 C r {\displaystyle C^{r}} vector fields (Leslie 1967). Over a compact subset of M {\displaystyle M} , this follows by fixing a Riemannian metric on M {\displaystyle M} and using the exponential map for that metric. If r {\displaystyle r} is finite and the manifold is compact, the space of vector fields is a Banach space. Moreover, the transition maps from one chart of this atlas to another are smooth, making the diffeomorphism group into a Banach manifold with smooth right translations; left translations and inversion are only continuous. If r = ∞ {\displaystyle r=\infty } , the space of vector fields is a Fréchet space. Moreover, the transition maps are smooth, making the diffeomorphism group into a Fréchet manifold and even into a regular Fréchet Lie group. If the manifold is σ {\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).
Lie algebra The Lie algebra of the diffeomorphism group of M {\displaystyle M} consists of all vector fields on M {\displaystyle M} equipped with the Lie bracket of vector fields. Somewhat formally, this is seen by making a small change to the coordinate x {\displaystyle x} at each point in space:
x μ ↦ x μ + ε h μ ( x ) {\displaystyle x^{\mu }\mapsto x^{\mu }+\varepsilon h^{\mu }(x)}
so the infinitesimal generators are the vector fields
L h = h μ ( x ) ∂ ∂ x μ . {\displaystyle L_{h}=h^{\mu }(x){\frac {\partial }{\partial x^{\mu }}}.}
Examples When M = G {\displaystyle M=G} is a Lie group, there is a natural inclusion of G {\displaystyle G} in its own diffeomorphism group via left-translation. Let Diff ( G ) {\displaystyle {\text{Diff}}(G)} denote the diffeomorphism group of G {\displaystyle G} , then there is a splitting Diff ( G ) ≃ G × Diff ( G , e ) {\displaystyle {\text{Diff}}(G)\simeq G\times {\text{Diff}}(G,e)} , where Diff ( G , e ) {\displaystyle {\text{Diff}}(G,e)} is the subgroup of Diff ( G ) {\displaystyle {\text{Diff}}(G)} that fixes the identity element of the group. The diffeomorphism group of Euclidean space R n {\displaystyle \mathbb {R} ^{n}} consists of two components, consisting of the orientation-preserving and orientation-reversing diffeomorphisms. In fact, the general linear group is a deformation retract of the subgroup Diff ( R n , 0 ) {\displaystyle {\text{Diff}}(\mathbb {R} ^{n},0)} of diffeomorphisms fixing the origin under the map f ( x ) → f ( t x ) / t , t ∈ ( 0 , 1 ] {\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 finite set of points, the diffeomorphism group is simply the symmetric group. Similarly, if M {\displaystyle M} is any manifold there is a group extension 0 → Diff 0 ( M ) → Diff ( M ) → Σ ( π 0 ( M ) ) {\displaystyle 0\to {\text{Diff}}_{0}(M)\to {\text{Diff}}(M)\to \Sigma (\pi _{0}(M))} . Here Diff 0 ( M ) {\displaystyle {\text{Diff}}_{0}(M)} is the subgroup of Diff ( M ) {\displaystyle {\text{Diff}}(M)} that preserves all the components of M {\displaystyle M} , and Σ ( π 0 ( M ) ) {\displaystyle \Sigma (\pi _{0}(M))} is the permutation group of the set π 0 ( M ) {\displaystyle \pi _{0}(M)} (the components of M {\displaystyle M} ). Moreover, the image of the map Diff ( M ) → Σ ( π 0 ( M ) ) {\displaystyle {\text{Diff}}(M)\to \Sigma (\pi _{0}(M))} is the bijections of π 0 ( M ) {\displaystyle \pi _{0}(M)} that preserve diffeomorphism classes.
Transitivity For a connected manifold M {\displaystyle M} , the diffeomorphism group acts transitively on M {\displaystyle M} . More generally, the diffeomorphism group acts transitively on the configuration space C k M {\displaystyle C_{k}M} . If M {\displaystyle M} is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space F k M {\displaystyle F_{k}M} and the action on M {\displayst