In mathematics, a submanifold of a manifold is a subset 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.

Immersed submanifolds

An immersed submanifold of an immersion; 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.

be an injection (one-to-one), in which we call it an injective immersion, and define an immersed submanifold together with a topology and differential structure is a diffeomorphism,, in the subset topology.

the image is a diffeomorphism. It follows that immersed submanifolds are precisely the images of injective immersions.

The submanifold topology on an immersed submanifold need not be the subspace topology. 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.

Embedded submanifolds

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

Given any embedding naturally has the structure of an embedded submanifold. That is, embedded submanifolds are precisely the images of embeddings.

there exists a chart-dimensional plane form an atlas.

Alexander's theorem and the Jordan–Schoenflies theorem 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.

. 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

, the tangent space can naturally be thought of as a linear subspace. This follows from the fact that the inclusion map is an immersion and provides an injection

Suppose S is closed is an embedded submanifold which is also a closed subset is closed if and only if it is a proper map (i.e. inverse images of compact sets is called a closed embedded submanifold. 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. This point of view is equivalent to the usual, abstract approach, because, by the Whitney embedding theorem, any second-countable.

See also

• Subspace (mathematics)

Notes

References