Diffeomorphism

Given two manifolds ${\displaystyle M}$ and ${\displaystyle N}$, a differentiable map ${\displaystyle f\colon M\rightarrow N}$ is called a diffeomorphism if it is a bijection and its inverse ${\displaystyle f^{-1}\colon N\rightarrow M}$ 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 ${\displaystyle N}$ are diffeomorphic (usually denoted ${\displaystyle M\simeq N}$) if there is a diffeomorphism ${\displaystyle f}$ from ${\displaystyle M}$ to ${\displaystyle N}$. They 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 a subset ${\displaystyle X}$ 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 a neighborhood ${\displaystyle U\subset M}$ of ${\displaystyle p}$ and a smooth function ${\displaystyle g:U\to N}$ such that the restrictions 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.

Hadamard-Caccioppoli Theorem[1]

If ${\displaystyle U}$, ${\displaystyle V}$ are connected open subsets of ${\displaystyle \mathbb {R} ^{n}}$ such that ${\displaystyle V}$ is simply connected, a differentiable map ${\displaystyle f:U\to V}$ is a diffeomorphism if it is proper and if the differential ${\displaystyle Df_{x}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ is bijective (and hence a linear isomorphism) at each point ${\displaystyle x}$ in ${\displaystyle U}$.

First remark

It is essential for ${\displaystyle V}$ to be simply 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 the complex square function

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

Then ${\displaystyle f}$ is surjective and it satisfies

${\displaystyle \det Df_{x}=4(x^{2}+y^{2})\neq 0.}$

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

Second remark

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

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

is a linear map, it has a well-defined inverse if and only if ${\displaystyle Df_{x}}$ is a bijection. The matrix representation of ${\displaystyle Df_{x}}$ is the ${\displaystyle n\times n}$ matrix of first-order partial derivatives whose entry in the ${\displaystyle i}$-th row and ${\displaystyle j}$-th column is ${\displaystyle \partial f_{i}/\partial x_{j}}$. This so-called Jacobian matrix is often used for explicit computations.

Third remark

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

Fourth remark

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

Fifth remark

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

Sixth remark

A differentiable bijection is not 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 a homeomorphism that is not a diffeomorphism.

Seventh remark

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 be differentiable; for a homeomorphism, ${\displaystyle f}$ and its inverse need only be continuous. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.

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

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:

${\displaystyle J_{f}={\begin{pmatrix}2x&3y^{2}\\2x&-3y^{2}\end{pmatrix}}.}$

The Jacobian matrix has zero determinant 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 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:

${\displaystyle J_{g}(0,0)={\begin{pmatrix}a_{1,0}&a_{0,1}\\b_{1,0}&b_{0,1}\end{pmatrix}}.}$

We see that g is a local diffeomorphism at 0 if, and only if,

${\displaystyle a_{1,0}b_{0,1}-a_{0,1}b_{1,0}\neq 0,}$

i.e. the linear terms in the components of g are linearly independent as polynomials.

• Let

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

We can calculate the Jacobian matrix:

${\displaystyle J_{h}={\begin{pmatrix}2x\cos(x^{2}+y^{2})&2y\cos(x^{2}+y^{2})\\-2x\sin(x^{2}+y^{2})&-2y\sin(x^{2}+y^{2})\end{pmatrix}}.}$

The Jacobian matrix has zero determinant everywhere! In fact we see that the image of h is the unit circle.

Let ${\displaystyle M}$ be a differentiable manifold that is second-countable and Hausdorff. The diffeomorphism group of ${\displaystyle M}$ is the group 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 not locally compact.

Since every diffeomorphism is a homeomorphism, given a pair of manifolds which are diffeomorphic to each other they are in particular homeomorphic 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 by John Milnor in dimension 7. He constructed a smooth 7-dimensional manifold (called now Milnor'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 a fiber bundle over the 4-sphere with the 3-sphere as the fiber).

More unusual phenomena occur for 4-manifolds. In the early 1980s, a combination of results due to Simon Donaldson and Michael Freedman led to the discovery of exotic ${\displaystyle \mathbb {R} ^{4}}$: there are uncountably 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 not embed smoothly in ${\displaystyle \mathbb {R} ^{4}}$.

• Anosov diffeomorphism such as Arnold's cat map
• Diffeo anomaly also known as a gravitational anomaly, a type anomaly in quantum 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 on Rⁿ.", 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