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 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 manifolds of class ''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 map ; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (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 immersion, and define an immersed submanifold to be the image subset ''S'' together with a

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

and differential structure 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 tw ...

: 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 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 inherited from ''M''. In general, it will be finer than the subspace topology (i.e. have more open sets). 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 subgroups are naturally immersed submanifolds. They also appear in the study of

foliations Foliation may refer to: * Foliation, a geometric device used to study manifolds * Foliation (geology), a property of certain rocks * A pagination system in book production * Vernation, the growth and arrangement of leaves * In architecture, an orn ...

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. That is, the submanifold topology on ''S'' is the same as the subspace topology. Given any embedding 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 containing ''p'' such that is the intersection of a ''k''-dimensional plane 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 geograp ...

for the differential structure on ''S''. Alexander's theorem 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 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 arcs and wild knots.

Properties

Given any immersed submanifold ''S'' of ''M'', the tangent space to a point ''p'' in ''S'' can naturally be thought of as a

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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 then ''S'' is actually an embedded submanifold of ''M''. Conversely, if ''S'' is an embedded submanifold which is also a closed subset then the inclusion map is closed. The inclusion map ''i'' : ''S'' → ''M'' is closed if and only if it is a proper map (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", ...

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, 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