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

, 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

. 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 ...

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 :

i_: T_p S \to T_p M.

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

* * * * * * {{Authority control Differential topology Manifolds