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 ...
, a submanifold of a
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 ...
''M'' is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
''S'' which itself has the structure of a manifold, and for which the
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iot ...
satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.
Formal definition
In the following we assume all manifolds are
differentiable 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 of
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differentl ...
''C''
''r'' for a fixed , and all morphisms are differentiable of class ''C''
''r''.
Immersed submanifolds
An immersed submanifold of a manifold ''M'' is the image ''S'' 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 ...
map ; in general this image will not be a submanifold as a subset, and an immersion map need not even be
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
(one-to-one) – it can have self-intersections.
More narrowly, one can require that the map be an injection (one-to-one), in which we call it an
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
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 ...
, and define an immersed submanifold to be the image subset ''S'' together with a
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and
differential 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 diff ...
such that ''S'' is a manifold and the inclusion ''f'' 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 ...
: this is just the topology on ''N,'' which in general will not agree with the subset topology: in general the subset ''S'' is not a submanifold of ''M,'' in the subset topology.
Given any injective immersion 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 ''N'' in ''M'' can be uniquely given the structure of an immersed submanifold so that 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 ...
. It follows that immersed submanifolds are precisely the images of injective immersions.
The submanifold topology on an immersed submanifold need not be the
relative topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
inherited from ''M''. In general, it will be
finer than the subspace topology (i.e. have more
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 ...
s).
Immersed submanifolds occur in the theory of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s where
Lie subgroup
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
s are naturally immersed submanifolds. They also appear in the study of
foliations where immersed submanifolds provide the right context to prove the
Frobenius theorem.
Embedded submanifolds
An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a
topological embedding
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 giv ...
. That is, the submanifold topology on ''S'' is the same as the subspace topology.
Given any
embedding
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 gi ...
of a manifold ''N'' in ''M'' the image ''f''(''N'') naturally has the structure of an embedded submanifold. That is, embedded submanifolds are precisely the images of embeddings.
There is an intrinsic definition of an embedded submanifold which is often useful. Let ''M'' be an ''n''-dimensional manifold, and let ''k'' be an integer such that . A ''k''-dimensional embedded submanifold of ''M'' is a subset such that for every point there exists a
chart
A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabu ...
containing ''p'' such that is the intersection of a ''k''-dimensional
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
with ''φ''(''U''). The pairs form an
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geographic ...
for the differential structure on ''S''.
Alexander's theorem
In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theorem is named after James Waddell Alexander ...
and the
Jordan–Schoenflies theorem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Moritz Schoenflies, Arthur Schoenflies. For Camille Jordan, Jordan curves in the Plane (geometry), plane it ...
are good examples of smooth embeddings.
Other variations
There are some other variations of submanifolds used in the literature. A
neat submanifold In differential topology, an area of mathematics, a neat submanifold of a manifold with boundary is a kind of "well-behaved" submanifold.
To define this more precisely, first let
:M be a manifold with boundary, and
:A be a submanifold of M.
Then ...
is a manifold whose boundary agrees with the boundary of the entire manifold. Sharpe (1997) defines a type of submanifold which lies somewhere between an embedded submanifold and an immersed submanifold.
Many authors define topological submanifolds also. These are the same as ''C''
''r'' submanifolds with .
[. ] An embedded topological submanifold is not necessarily regular in the sense of the existence of a local chart at each point extending the embedding. Counterexamples include
wild arc
In geometric topology, a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc to a straight line segment. found the fi ...
s and
wild knot
Wild, wild, wilds or wild may refer to:
Common meanings
* Wild animal
* Wilderness, a wild natural environment
* Wildness, the quality of being wild or untamed
Art, media and entertainment Film and television
* ''Wild'' (2014 film), a 2014 Am ...
s.
Properties
Given any immersed submanifold ''S'' of ''M'', the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
to a point ''p'' in ''S'' can naturally be thought of as a
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
of the tangent space to ''p'' in ''M''. This follows from the fact that the inclusion map is an immersion and provides an injection
:
Suppose ''S'' is an immersed submanifold of ''M''. If the inclusion map is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
then ''S'' is actually an embedded submanifold of ''M''. Conversely, if ''S'' is an embedded submanifold which is also a
closed subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
then the inclusion map is closed. The inclusion map ''i'' : ''S'' → ''M'' is closed if and only if it is a
proper map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
Definition
There are several competing definitio ...
(i.e. inverse images of
compact set
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", i ...
s are compact). If ''i'' is closed then ''S'' is called a closed embedded submanifold of ''M''. Closed embedded submanifolds form the nicest class of submanifolds.
Submanifolds of real coordinate space
Smooth manifolds are sometimes ''defined'' as embedded submanifolds of
real coordinate space
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
R
''n'', for some ''n''. This point of view is equivalent to the usual, abstract approach, because, by the
Whitney embedding theorem
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:
*The strong Whitney embedding theorem states that any differentiable manifold, smooth real numbers, real -dimension (math ...
, any
second-countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
smooth (abstract) ''m''-manifold can be smoothly embedded in R
2''m''.
Notes
References
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Differential topology
Manifolds