In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, a local diffeomorphism is intuitively a
map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
between
Smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s that preserves the local
differentiable structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for dif ...
. The formal definition of a local diffeomorphism is given below.
Formal definition
Let
and
be
differentiable manifolds. A
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is a local diffeomorphism, if for each point
there exists an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
containing
such that
is open in
and
is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
.
A local diffeomorphism is a special case of an
immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
where the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of
under
locally has the differentiable structure of a
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
of
Then
and
may have a lower dimension than
Characterizations
A map is a local diffeomorphism if and only if it is a smooth
immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
(smooth local embedding) and an
open map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, a ...
.
The
inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at ...
implies that a smooth map
is a local diffeomorphism if and only if the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is a
linear isomorphism
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
for all points
This implies that
and
must have the same dimension.
A map
between two connected manifolds of equal dimension (
) is a local diffeomorphism if and only if it is a smooth
immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
(smooth local embedding), or equivalently, if and only if it is a smooth
submersion.
This is because every smooth immersion is a
locally injective function
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
while
invariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n.
It states:
:If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomorph ...
guarantees that any continuous injective function between manifolds of equal dimensions is necessarily an open map.
Discussion
For instance, even though all manifolds look locally the same (as
for some
) in the topological sense, it is natural to ask whether their differentiable structures behave in the same manner locally. For example, one can impose two different
differentiable structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for dif ...
s on
that make
into a differentiable manifold, but both structures are not locally diffeomorphic (see below). Although local 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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. For example, there can be no global 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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
, the continuous image of a
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
is compact, the 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
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If f : X \to Y is a local homeomorphism, X is said to be an à ...
and therefore a
locally injective open map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, a ...
.
A local diffeomorphism has constant
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
of
Examples
A
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
is a
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
local diffeomorphism.
A
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
is a local diffeomorphism such that every point in the target has a
neighborhood that is by the map.
Local flow diffeomorphisms
See also
*
*
*
*
*
*
References
* .
{{DEFAULTSORT:Local Diffeomorphism
Theory of continuous functions
Diffeomorphisms
Functions and mappings
Inverse functions