Diffeomorphism Explained
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
and
, a differentiable map
is a
diffeomorphism if it is a
bijection and its inverse
is differentiable as well. If these functions are
times continuously differentiable,
is called a
-diffeomorphism.
Two manifolds
and
are
diffeomorphic (usually denoted
) if there is a diffeomorphism
from
to
. Two
-differentiable manifolds are
-diffeomorphic if there is an
times continuously differentiable bijective map between them whose inverse is also
times continuously differentiable.
Diffeomorphisms of subsets of manifolds
of a manifold
and a subset
of a manifold
, a function
is said to be smooth if for all
in
there is a
neighborhood
of
and a smooth function
such that the
restrictions agree:
(note that
is an extension of
). The function
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:[1]
If
,
are
connected open subsets of
such that
is
simply connected, a differentiable map
is a diffeomorphism if it is
proper and if the
differential
is bijective (and hence a
linear isomorphism) at each point
in
.
Some remarks:
It is essential for
to be
simply connected for the function
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
\begin{cases}
f:\R2\setminus\{(0,0)\}\to\R2\setminus\{(0,0)\}\\
(x,y)\mapsto(x2-y2,2xy).
\end{cases}
Then
is
surjective and it satisfies
Thus, though
is bijective at each point,
is not invertible because it fails to be
injective (e.g.
).
Since the differential at a point (for a differentiable function)
is a
linear map, it has a well-defined inverse if and only if
is a bijection. The
matrix representation of
is the
matrix of first-order
partial derivatives whose entry in the
-th row and
-th column is
. This so-called
Jacobian matrix is often used for explicit computations.
Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine
going from dimension
to dimension
. If
then
could never be surjective, and if
then
could never be injective. In both cases, therefore,
fails to be a bijection.
If
is a bijection at
then
is said to be a
local diffeomorphism (since, by continuity,
will also be bijective for all
sufficiently close to
).
Given a smooth map from dimension
to dimension
, if
(or, locally,
) is surjective,
is said to be a
submersion (or, locally, a "local submersion"); and if
(or, locally,
) is injective,
is said to be an
immersion (or, locally, a "local immersion").
A differentiable bijection is not necessarily a diffeomorphism.
, for example, is not a diffeomorphism from
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
is a map between differentiable manifolds, a diffeomorphic
is a stronger condition than a homeomorphic
. For a diffeomorphism,
and its inverse need to be differentiable; for a homeomorphism,
and its inverse need only be
continuous. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.
is a diffeomorphism if, in coordinate charts, it satisfies the definition above. More precisely: Pick any cover of
by compatible coordinate charts and do the same for
. Let
and
be charts on, respectively,
and
, with
and
as, respectively, the images of
and
. The map
is then a diffeomorphism as in the definition above, whenever
f(\phi-1(U))\subseteq\psi-1(V)
.
Examples
Since any manifold can be locally parametrised, we can consider some explicit maps from
into
.
f(x,y)=\left(x2+y3,x2-y3\right).
We can calculate the Jacobian matrix:
Jf=\begin{pmatrix}2x&3y2\ 2x&-3y2\end{pmatrix}.
The Jacobian matrix has zero determinant if and only if
. We see that
could only be a diffeomorphism away from the
-axis and the
-axis. However,
is not bijective since
, and thus it cannot be a diffeomorphism.
g(x,y)=\left(a0+a1,0x+a0,1y+ … , b0+b1,0x+b0,1y+ … \right)
where the
and
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:
Jg(0,0)=\begin{pmatrix}a1,0&a0,1\ b1,0&b0,1\end{pmatrix}.
We see that g is a local diffeomorphism at 0 if, and only if,
i.e. the linear terms in the components of g are linearly independent as polynomials.
h(x,y)=\left(\sin(x2+y2),\cos(x2+y2)\right).
We can calculate the Jacobian matrix:
Jh=\begin{pmatrix}2x\cos(x2+y2)&2y\cos(x2+y2)\ -2x\sin(x2+y2)&-2y\sin(x2+y2)\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
between two
surfaces
and
has a Jacobian matrix
that is an
invertible matrix. In fact, it is required that for
in
, there is a
neighborhood of
in which the Jacobian
stays
non-singular. Suppose that in a chart of the surface,
The total differential of u is
, and similarly for
v.Then the image
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
be a differentiable manifold that is
second-countable and
Hausdorff. The
diffeomorphism group of
is the
group of all
diffeomorphisms of
to itself, denoted by
or, when
is understood,
. This is a "large" group, in the sense that—provided
is not zero-dimensional—it is not
locally compact.
Topology
The diffeomorphism group has two natural topologies: weak and strong . 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
, the weak topology is the topology induced by the family of metrics
dK(f,g)=\sup\nolimitsx\ind(f(x),g(x))+\sum\nolimits1\le\sup\nolimitsx\in\left\|Dpf(x)-Dpg(x)\right\|
as
varies over compact subsets of
. Indeed, since
is
-compact, there is a sequence of compact subsets
whose
union is
. Then:
d(f,g)=\sum\nolimitsn2-n
.
The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of
vector fields . Over a compact subset of
, this follows by fixing a Riemannian metric on
and using the
exponential map for that metric. If
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
, 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
-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 .
Lie algebra
The Lie algebra of the diffeomorphism group of
consists of all
vector fields on
equipped with the
Lie bracket of vector fields. Somewhat formally, this is seen by making a small change to the coordinate
at each point in space:
x\mu\mapstox\mu+\varepsilonh\mu(x)
so the infinitesimal generators are the vector fields
Examples
is a
Lie group, there is a natural inclusion of
in its own diffeomorphism group via left-translation. Let
denote the diffeomorphism group of
, then there is a splitting
Diff(G)\simeqG x Diff(G,e)
, where
is the
subgroup of
that fixes the
identity element of the group.
- The diffeomorphism group of Euclidean space
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
of diffeomorphisms fixing the origin under the map
. 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
is any manifold there is a
group extension 0\toDiff0(M)\toDiff(M)\to\Sigma(\pi0(M))
. Here
is the subgroup of
that preserves all the components of
, and
is the permutation group of the set
(the components of
). Moreover, the image of the map
Diff(M)\to\Sigma(\pi0(M))
is the bijections of
that preserve diffeomorphism classes.
Transitivity
For a connected manifold
, the diffeomorphism group
acts transitively on
. More generally, the diffeomorphism group acts transitively on the
configuration space
. If
is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space
and the action on
is multiply transitive .
Extensions of diffeomorphisms
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.
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
of the reals satisfying
; 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 the
Alexander trick). Moreover, the diffeomorphism group of the circle has the homotopy-type of the
orthogonal group
.
The corresponding extension problem for diffeomorphisms of higher-dimensional spheres
was much studied in the 1950s and 1960s, with notable contributions from
René Thom,
John Milnor and
Stephen Smale. An obstruction to such extensions is given by the finite
abelian group
, the "group of twisted spheres", defined as the
quotient of the abelian component group of the diffeomorphism group by the subgroup of classes extending to diffeomorphisms of the ball
.
Connectedness
For manifolds, the diffeomorphism group is usually not connected. Its component group is called the mapping class group. 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). Max Dehn and Jakob Nielsen showed that it can be identified with the outer automorphism group of the fundamental group of the surface.
, the mapping class group is simply the
modular group
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. Smale
conjectured that if
is an oriented smooth closed manifold, the
identity component of the group of orientation-preserving diffeomorphisms is
simple. This had first been proved for a product of circles by Michel Herman; it was proved in full generality by Thurston.
Homotopy types
- The diffeomorphism group of
has the homotopy-type of the subgroup
. This was proven by Steve Smale.
[2] - The diffeomorphism group of the torus has the homotopy-type of its linear automorphisms:
.
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 finite fundamental groups).
- The homotopy-type of diffeomorphism groups of
-manifolds for
are poorly understood. For example, it is an open problem whether or not
has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided
,
does not have the homotopy-type of a finite
CW-complex.
Homeomorphism and diffeomorphism
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).
: there are
uncountably many pairwise non-diffeomorphic open subsets of
each of which is homeomorphic to
, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to
that do not embed smoothly in
.
See also
References
- Book: 978-1-4614-5980-4. Krantz. Steven G.. Parks. Harold R.. The implicit function theorem: history, theory, and applications. Boston. Modern Birkhäuser classics. 2013.
- Chaudhuri . Shyamoli . Kawai . Hikaru . Tye . S.-H. Henry . Path-integral formulation of closed strings . Physical Review D . 36 . 4 . 1987-08-15 . 0556-2821 . 10.1103/physrevd.36.1148 . 1148–1168. 9958280 . 1987PhRvD..36.1148C . 41709882 . https://web.archive.org/web/20180721010540/http://repository.ust.hk/ir/bitstream/1783.1-49255/1/PhysRevD.36.1148.pdf . 2018-07-21 . live .
Notes and References
- Book: Steven G. Krantz . Harold R. Parks . The implicit function theorem: history, theory, and applications . 2013 . 978-1-4614-5980-4 . Theorem 6.2.4. Springer .
- Smale . 1959 . Diffeomorphisms of the 2-sphere . Proc. Amer. Math. Soc. . 10 . 4. 621–626 . 10.1090/s0002-9939-1959-0112149-8. free .