# Diffeomorphism

A diffeomorphism is a map between differentiable manifolds that is a bijection and is smooth in both directions: the map and its inverse are both infinitely differentiable. Two manifolds related by a diffeomorphism are called diffeomorphic, and from the point of view of differential topology they have the same properties; classification of manifolds up to diffeomorphism is finer than classification up to homeomorphism, except in small dimensions.<sup>[1](https://encyclopediaofmath.org/wiki/Diffeomorphism)</sup><sup> • </sup><sup>[2](https://math.colorado.edu/~casa/teaching/26fall/4230/lecture_notes/index.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | A smooth bijection whose inverse is also smooth; equivalently, a differentiable homeomorphism with differentiable inverse<sup>[2](https://math.colorado.edu/~casa/teaching/26fall/4230/lecture_notes/index.html)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Diffeomorphism)</sup> |
| Derivative | If \( f \colon M \to N \) is a diffeomorphism, the derivative \( df_p \colon T_p M \to T_{f(p)} N \) is an isomorphism for every \( p \)<sup>[3](https://gt.postech.ac.kr/~jccha/intro-geom-top-lecture-12-2023-spring/)</sup> |
| Local test | Inverse function theorem: if \( df(x) \) is an isomorphism for a \( C^1 \) map between open sets of \( \mathbb{R}^n \), the map is a local diffeomorphism near \( x \)<sup>[4](https://www.math.ucla.edu/~honda/math225a/revised%20course%20notes.pdf)</sup> |
| Integration | In the change-of-variables theorem, a diffeomorphism \( F \colon U \to V \) between open subsets of \( \mathbb{R}^n \) transforms integrals by the factor \( \lvert \det DF \rvert \)<sup>[5](https://math.mit.edu/~larsh/teaching/F2007/handouts/changeofvariables.pdf)</sup> |
| Exotic spheres | The 7-sphere carries exactly 28 distinct differentiable structures (Milnor and Kervaire), and \( S^{31} \) carries 16,931,177<sup>[6](https://mathshistory.st-andrews.ac.uk/Extras/Milnor_awards/)</sup> |
| Physics | In general relativity, diffeomorphisms are the gauge symmetry, the redundancy of description of the theory<sup>[7](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)</sup> |

## Definition and first consequences

For subsets of [Euclidean space](https://www.edgepedia.org/euclidean-space), a bijection \( f \colon X \to Y \), with \( X \subset \mathbb{R}^n \) and \( Y \subset \mathbb{R}^m \), is a diffeomorphism when both \( f \) and its inverse \( f^{-1} \colon Y \to X \) are smooth; one then says that \( X \) and \( Y \) are diffeomorphic subsets.<sup>[8](https://www.studocu.com/en-us/document/the-university-of-texas-at-tyler/partial-differential-equations/rlf-notes/24962523)</sup> For smooth manifolds the definition is the same: a smooth map \( f \colon M \to N \) is a diffeomorphism if there exists a smooth inverse \( f^{-1} \colon N \to M \).<sup>[4](https://www.math.ucla.edu/~honda/math225a/revised%20course%20notes.pdf)</sup> [Smoothness](https://www.edgepedia.org/smoothness) of a map between manifolds is itself defined through coordinate charts, so the notion reduces locally to smoothness of maps between open sets of Euclidean spaces.<sup>[4](https://www.math.ucla.edu/~honda/math225a/revised%20course%20notes.pdf)</sup>

The smoothness requirement has immediate consequences. Composing \( f \) with its inverse and applying the chain rule gives \( D(f^{-1})(y) \cdot Df(x) = I \), so the matrix of partial derivatives \( Df(x) \) is invertible at every point of the domain.<sup>[5](https://math.mit.edu/~larsh/teaching/F2007/handouts/changeofvariables.pdf)</sup> Equivalently, if \( f \) is a diffeomorphism of smooth manifolds, the derivative \( df_p \) is an isomorphism on tangent spaces for all \( p \).<sup>[3](https://gt.postech.ac.kr/~jccha/intro-geom-top-lecture-12-2023-spring/)</sup> A diffeomorphism is in particular a homeomorphism, and one textbook definition states it as a homeomorphism in which both the mapping and its inverse are also differentiable.<sup>[9](https://www.studocu.com/en-gb/document/university-of-dundee/differential-geometry/lecture-notes-ma32007-differential-geometry/14475736)</sup>

A map is a local diffeomorphism at a point \( x \) if some open neighborhood of \( x \) is carried diffeomorphically onto an open set.<sup>[10](https://citeseerx.ist.psu.edu/document?doi=9490755a4c0341defc70e09fe86d2d2a53e364bb&repid=rep1&type=pdf)</sup> A bijective local diffeomorphism is a diffeomorphism onto its image, since smoothness of the inverse holds locally at every point.<sup>[11](https://math.stackexchange.com/questions/2630351/bijective-local-diffeomorphism-is-a-diffeomorphism)</sup>

## The inverse function theorem

The inverse function theorem is the standard tool for producing diffeomorphisms. In its basic form, if \( f \colon U \to V \) is a \( C^1 \) map between open sets of \( \mathbb{R}^n \) and the derivative \( df(x) \colon \mathbb{R}^n \to \mathbb{R}^n \) is an isomorphism, then \( f \) is a local diffeomorphism near \( x \): there are open sets \( U_x \ni x \) and \( V_{f(x)} \ni f(x) \) such that \( f \rvert_{U_x} \colon U_x \to V_{f(x)} \) is a diffeomorphism.<sup>[4](https://www.math.ucla.edu/~honda/math225a/revised%20course%20notes.pdf)</sup> The theorem is described in those course notes as the most important basic theorem in differential geometry, moving from an isomorphism in the linear category to a local diffeomorphism in the differentiable category.

The converse is almost true globally. If \( F \colon U \to \mathbb{R}^n \) is \( C^1 \), one-to-one, and has invertible derivative \( DF(x) \) at every point, then \( V = F(U) \) is open and \( F \colon U \to V \) is a diffeomorphism.<sup>[5](https://math.mit.edu/~larsh/teaching/F2007/handouts/changeofvariables.pdf)</sup> Local invertibility alone does not give a global inverse; additional conditions such as properness appear in global inverse function theorems in the tradition of Hadamard.<sup>[12](https://arxiv.org/abs/1305.5930)</sup>

## Change of variables

In multivariable calculus, the change of variables theorem is stated in terms of diffeomorphisms. For a diffeomorphism \( F \colon U \to V \) between open subsets of \( \mathbb{R}^2 \) and bounded subsets \( D^* \subset U \), \( D = F(D^*) \), the theorem gives

\[ \iint_D f(x,y) \, dx \, dy = \iint_{D^*} f(F(u,v)) \, \lvert \det DF(u,v) \rvert \, du \, dv, \]

with the entirely analogous formula in three variables.<sup>[5](https://math.mit.edu/~larsh/teaching/F2007/handouts/changeofvariables.pdf)</sup> The determinant \( \det DF \), the Jacobian, measures how the map distorts volume, and its absolute value corrects the integral for that distortion.<sup>[5](https://math.mit.edu/~larsh/teaching/F2007/handouts/changeofvariables.pdf)</sup> The same framework extends to integration of differential forms on manifolds, where the change-of-variables formula for a diffeomorphism \( F \) and a compact set \( K \) reads \( \int_K f \, dV = \int_{F^{-1}(K)} (f \circ F) \det(dF) \).<sup>[13](https://www.math.brown.edu/reschwar/M114B/notes8.pdf)</sup>

## Diffeomorphism groups and mapping class groups

The diffeomorphisms of a manifold \( M \) to itself form a group under composition, denoted \( \mathrm{Diff}(M) \). Many classical families of transformations are groups of diffeomorphisms; older names such as contact transformations, canonical mappings, conformal mappings, and biholomorphic mappings describe diffeomorphisms preserving an additional structure, and the modern usage attaches an adjective, as in symplectic diffeomorphism.<sup>[1](https://encyclopediaofmath.org/wiki/Diffeomorphism)</sup>

For surfaces, the quotient \( \mathrm{Mod}(S_g) := \mathrm{Diff}(S_g) / \mathrm{Diff}_0(S_g) \) of the diffeomorphism group by the subgroup of diffeomorphisms isotopic to the identity is the mapping class group, isomorphic to the group \( \pi_0 \mathrm{Diff}(S_g) \) of path components.<sup>[14](https://nsalter.science.nd.edu/expository-notes/sbsurvey.pdf)</sup> For the torus this gives \( \mathrm{Mod}(T^2) \cong \mathrm{SL}_2(\mathbb{Z}) \).<sup>[14](https://nsalter.science.nd.edu/expository-notes/sbsurvey.pdf)</sup> The mapping class group classifies surface bundles over the circle built by the mapping torus construction, in which a diffeomorphism \( f \colon S \to S \) glues the two ends of \( [0,1] \times S \) together.<sup>[14](https://nsalter.science.nd.edu/expository-notes/sbsurvey.pdf)</sup>

The group \( \mathrm{Diff}(M) \) is also an object of study in its own right and in infinite-dimensional geometry. [Vladimir Arnold](https://www.edgepedia.org/vladimir-arnold)'s 1966 discovery that solutions of Euler's equations for an incompressible ideal fluid correspond to geodesics on the infinite-dimensional manifold of volume-preserving diffeomorphisms founded the field of geometric hydrodynamics.<sup>[15](https://arxiv.org/abs/1910.03301)</sup>

## Diffeomorphism versus homeomorphism: exotic structures

Every diffeomorphism is a homeomorphism, but the converse fails, and the failures define much of differential topology. A smooth structure on a topological manifold is an atlas of coordinate charts whose transition functions are smooth; two smooth structures are considered equivalent when there is a diffeomorphism relating them.<sup>[16](https://mathworld.wolfram.com/SmoothStructure.html)</sup> Manifolds that are homeomorphic but not diffeomorphic are said to carry exotic smooth structures.

**Milnor's exotic spheres.** In 1956 [John Milnor](https://www.edgepedia.org/john-milnor), an American topologist then at Princeton University, published "On manifolds homeomorphic to the 7-sphere" in the [Annals of Mathematics](https://www.edgepedia.org/annals-of-mathematics), exhibiting a smooth manifold homeomorphic but not diffeomorphic to the standard 7-sphere.<sup>[17](https://sites.math.rutgers.edu/~feehan/teaching/math866/milnor7sphere.pdf)</sup><sup> • </sup><sup>[18](https://www.sciencedirect.com/topics/mathematics/4-manifold)</sup> The discovery came as a complete surprise and marked the beginning of differential topology as a field.<sup>[6](https://mathshistory.st-andrews.ac.uk/Extras/Milnor_awards/)</sup> With Michel Kervaire, a Swiss mathematician, Milnor later classified distinct differentiable structures on spheres in dimensions at least 5: the 7-sphere carries exactly 28, and \( S^{31} \) carries 16,931,177.<sup>[6](https://mathshistory.st-andrews.ac.uk/Extras/Milnor_awards/)</sup><sup> • </sup><sup>[24](https://euromathsoc.org/magazine/articles/142)</sup> Homeomorphic but not diffeomorphic manifolds were subsequently found in all dimensions greater than 4, by Kervaire and Milnor (1963), Barden (1965), and Siebenmann (1971).<sup>[19](https://armj.math.stonybrook.edu/pdf-Springer-final/015-0007-5.pdf)</sup>

**Dimension four.** [Dimension](https://www.edgepedia.org/dimension) 4 behaves differently from every other dimension. On a fixed compact topological manifold of dimension other than 4 there are only finitely many non-diffeomorphic smooth structures.<sup>[20](https://web.ma.utexas.edu/users/perutz/PDF_files/Smooth4manifolds.pdf)</sup> For closed simply connected 4-manifolds, Freedman's 1982 theorem determines homeomorphism by the intersection form, while Donaldson's 1987 gauge-theoretic results showed that some smooth manifolds homeomorphic by those criteria are not diffeomorphic; all invariants used to distinguish smooth structures on such manifolds are rooted in analysis, counting solutions of nonlinear partial differential equations, chiefly the Donaldson and [Seiberg–Witten invariants](https://www.edgepedia.org/seiberg-witten-invariants).<sup>[19](https://armj.math.stonybrook.edu/pdf-Springer-final/015-0007-5.pdf)</sup> Recent work continues the theme: a 2026 Annals of Mathematics paper proves the existence of closed, aspherical, smooth 4-manifolds that are homeomorphic but not diffeomorphic, giving counterexamples to a smooth analog of the Borel conjecture in dimension four.<sup>[21](https://annals.math.princeton.edu/2026/204-2/p06)</sup>

## Diffeomorphisms in physics

In general relativity, spacetime is modeled as a manifold \( M \) carrying tensor fields such as the metric \( g \). If \( \phi \colon M \to N \) is a diffeomorphism, there is no way of distinguishing \( (M, g, F, \ldots) \) from \( (N, \phi_*(g), \phi_*(F), \ldots) \); the two describe the same physics, so diffeomorphisms are the gauge symmetry, the redundancy of description, of the theory.<sup>[7](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)</sup> A diffeomorphism \( \phi \colon M \to M \) is a symmetry transformation of a tensor field \( T \) precisely when \( \phi_*(T) = T \) everywhere.<sup>[7](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)</sup> Such diffeomorphisms are analogous to the gauge symmetries of Maxwell and [Yang–Mills theory](https://www.edgepedia.org/yang-mills-theory), and one may view them either as an active change, mapping one point to another, or as a passive change of coordinate charts labeling the same point.<sup>[22](https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S4.html)</sup> In the Hamiltonian formulation, the first-class constraints of the Einstein–Hilbert Lagrangian generate gauge transformations that lead directly to diffeomorphism invariance.<sup>[23](https://arxiv.org/abs/0808.2623)</sup>

## References

1. [Diffeomorphism, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Diffeomorphism)
2. [Differential Geometry of Curves and Surfaces lecture notes, University of Colorado Boulder](https://math.colorado.edu/~casa/teaching/26fall/4230/lecture_notes/index.html)
3. [Introduction to Geometric Topology, Lecture Note 12, Jae Choon Cha, POSTECH](https://gt.postech.ac.kr/~jccha/intro-geom-top-lecture-12-2023-spring/)
4. [Differential Geometry Course Notes, Ko Honda, UCLA](https://www.math.ucla.edu/~honda/math225a/revised%20course%20notes.pdf)
5. [18.022: Multivariable calculus — The change of variables theorem, MIT](https://math.mit.edu/~larsh/teaching/F2007/handouts/changeofvariables.pdf)
6. [Milnor awards, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Extras/Milnor_awards/)
7. [General Relativity lecture notes, H. S. Reall, University of Cambridge](https://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf)
8. [Differential Geometry Lecture Notes for RLF, University of Texas at Tyler (Studocu copy)](https://www.studocu.com/en-us/document/the-university-of-texas-at-tyler/partial-differential-equations/rlf-notes/24962523)
9. [Lecture Notes MA32007: Differential Geometry, University of Dundee (Studocu copy)](https://www.studocu.com/en-gb/document/university-of-dundee/differential-geometry/lecture-notes-ma32007-differential-geometry/14475736)
10. [Some Notes on Differential Geometry, CiteSeerX](https://citeseerx.ist.psu.edu/document?doi=9490755a4c0341defc70e09fe86d2d2a53e364bb&repid=rep1&type=pdf)
11. [Bijective local diffeomorphism is a diffeomorphism, Mathematics Stack Exchange](https://math.stackexchange.com/questions/2630351/bijective-local-diffeomorphism-is-a-diffeomorphism)
12. [On Hadamard's global inverse function theorem, arXiv](https://arxiv.org/abs/1305.5930)
13. [The Change of Variables Formula, Rich Schwartz, Brown University](https://www.math.brown.edu/reschwar/M114B/notes8.pdf)
14. [Surface bundles, mapping class groups, moduli spaces, and cohomology, Nick Salter and Bena Tshishiku](https://nsalter.science.nd.edu/expository-notes/sbsurvey.pdf)
15. [Geometric Hydrodynamics: from Euler, to Poincaré, to Arnold, arXiv](https://arxiv.org/abs/1910.03301)
16. [Smooth Structure, Wolfram MathWorld](https://mathworld.wolfram.com/SmoothStructure.html)
17. [John Milnor, On Manifolds Homeomorphic to the 7-Sphere, Annals of Mathematics 64 (1956)](https://sites.math.rutgers.edu/~feehan/teaching/math866/milnor7sphere.pdf)
18. [4-Manifold overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/4-manifold)
19. [O. Viro, Knot Space of a 4-Manifold, Arnold Mathematical Journal](https://armj.math.stonybrook.edu/pdf-Springer-final/015-0007-5.pdf)
20. [Smooth 4-manifolds notes, Tim Perutz, University of Texas at Austin](https://web.ma.utexas.edu/users/perutz/PDF_files/Smooth4manifolds.pdf)
21. [Exotic aspherical 4-manifolds, Annals of Mathematics 204 (2026)](https://annals.math.princeton.edu/2026/204-2/p06)
22. [The Einstein Equations, General Relativity lecture notes, David Tong, University of Cambridge](https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S4.html)
23. [Diffeomorphism Invariance in the Hamiltonian formulation of General Relativity, arXiv](https://arxiv.org/abs/0808.2623)
24. [Stable homotopy groups, EMS Magazine](https://euromathsoc.org/magazine/articles/142)

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