Diffeomorphism
times continuously differentiable bijective map between them whose inverse is also
Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions.
to be globally invertible (under the sole condition that its derivative be a bijective map at each point).
In mechanics, a stress-induced transformation is called a deformation and may be described by a diffeomorphism.
is a linear transformation, fixing the origin, and expressible as the action of a complex number of a particular type.
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.
The diffeomorphism group has two natural topologies: weak and strong (Hirsch 1997).
When the manifold is not compact, the strong topology captures the behavior of functions "at infinity" and is not metrizable.
Then: The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of
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.
-compact and not compact the full diffeomorphism group is not locally contractible for any of the two topologies.
at each point in space: so the infinitesimal generators are the vector fields For a connected manifold
More generally, the diffeomorphism group acts transitively on the configuration space
is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space
In 1926, Tibor Radó asked whether the harmonic extension of any homeomorphism or diffeomorphism of the unit circle to the unit disc yields a diffeomorphism on the open disc.
An elegant proof was provided shortly afterwards by Hellmuth Kneser.
In 1945, Gustave Choquet, apparently unaware of this result, produced a completely different proof.
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).
was much studied in the 1950s and 1960s, with notable contributions from René Thom, John Milnor and Stephen Smale.
In dimension 2 (i.e. surfaces), the mapping class group is a finitely presented group generated by Dehn twists; this has been proved by Max Dehn, W. B. R. Lickorish, and Allen Hatcher).
William 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.
and the classification becomes classical in terms of elliptic, parabolic and hyperbolic matrices.
Thurston accomplished his classification by observing that the mapping class group acted naturally on a compactification of Teichmüller space; as this enlarged space was homeomorphic to a closed ball, the Brouwer fixed-point theorem became applicable.
is an oriented smooth closed manifold, the identity component of the group of orientation-preserving diffeomorphisms is simple.
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.
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.
In the early 1980s, a combination of results due to Simon Donaldson and Michael Freedman led to the discovery of exotic
, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to
