Local diffeomorphism explained
In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.
Formal definition
Let
and
be
differentiable manifolds. A
function
is a
local diffeomorphism if, for each point
, there exists an
open set
containing
such that the
image
is open in
and
is a
diffeomorphism.
. In this case, for each
, there exists an open set
containing
such that the image
is an embedded submanifold, and
is a diffeomorphism. Here
and
have the same dimension, which may be less than the dimension of
.
[1] Characterizations
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map.
The inverse function theorem implies that a smooth map
is a local diffeomorphism if and only if the
derivative
is a
linear isomorphism for all points
. This implies that
and
have the same dimension.
[2] It follows that a map
between two manifolds of equal dimension (
\operatorname{dim}X=\operatorname{dim}Y
) is a local diffeomorphism if and only if it is a smooth
immersion (smooth local embedding), or equivalently, if and only if it is a smooth
submersion. This is because, for any
, both
and
have the same dimension, thus
is a linear isomorphism if and only if it is injective, or equivalently, if and only if it is surjective.
[3] Here is an alternative argument for the case of an immersion: every smooth immersion is a locally injective function, while invariance of domain guarantees that any continuous injective function between manifolds of equal dimensions is necessarily an open map.
Discussion
All manifolds of the same dimension are "locally diffeomorphic," in the following sense: if
and
have the same dimension, and
and
, then there exist open neighbourhoods
of
and
of
and a diffeomorphism
. However, this map
need not extend to a smooth map defined on all of
, let alone extend to a local diffeomorphism. Thus the existence of a local diffeomorphism
is a stronger condition than "to be locally diffeomophic." Indeed, although locally defined diffeomorphisms preserve differentiable structure locally, one must be able to "patch up" these (local) diffeomorphisms to ensure that the domain is the entire smooth manifold.
For example, one can impose two different differentiable structures on
that each make
into a differentiable manifold, but both structures are not locally diffeomorphic (see Exotic
).
As another example, there can be no local diffeomorphism from the 2-sphere to Euclidean 2-space, although they do indeed have the same local differentiable structure. This is because all local diffeomorphisms are continuous, the continuous image of a compact space is compact, and the 2-sphere is compact whereas Euclidean 2-space is not.
Properties
If a local diffeomorphism between two manifolds exists then their dimensions must be equal. Every local diffeomorphism is also a local homeomorphism and therefore a locally injective open map. A local diffeomorphism has constant rank of
Examples
Local flow diffeomorphisms
See also: Flow (mathematics).
References
Notes and References
- Lee, Introduction to smooth manifolds, Proposition 5.22
- Lee, Introduction to smooth manifolds, Proposition 4.8
- Axler, Linear algebra done right, Theorem 3.21
