Submanifold

In mathematics, a submanifold of a manifold

which itself has the structure of a manifold, and for which the inclusion map

There are different types of submanifolds depending on exactly which properties are required.

, and all morphisms are differentiable of class

An immersed submanifold of a manifold

; 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.

[1] 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

together with a topology and differential structure such that

can be uniquely given the structure of an immersed submanifold so that

The submanifold topology on an immersed submanifold need not be the subspace topology inherited from

In general, it will be finer than the subspace topology (i.e. have more open sets).

Immersed submanifolds occur in the theory of Lie groups where Lie subgroups are naturally immersed submanifolds.

They also appear in the study of foliations where immersed submanifolds provide the right context to prove the Frobenius theorem.

An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a topological embedding.

naturally has the structure of an embedded submanifold.

There is an intrinsic definition of an embedded submanifold which is often useful.

form an atlas for the differential structure on

Alexander's theorem and the Jordan–Schoenflies theorem are good examples of smooth embeddings.

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.

Many authors define topological submanifolds also.

[3] An embedded topological submanifold is not necessarily regular in the sense of the existence of a local chart at each point extending the embedding.

, the tangent space to a point

can naturally be thought of as a linear subspace of the tangent space to

This follows from the fact that the inclusion map is an immersion and provides an injection Suppose S is an immersed submanifold of

is closed if and only if it is a proper map (i.e. inverse images of compact sets are compact).

is called a closed embedded submanifold of

Smooth manifolds are sometimes defined as embedded submanifolds of real coordinate space

This point of view is equivalent to the usual, abstract approach, because, by the Whitney embedding theorem, any second-countable smooth (abstract)