In mathematics, a submanifold of a manifoldM is a subsetS which itself has the structure of a manifold, and for which the inclusion mapS → M 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 classCr for a fixed r ≥ 1, and all morphisms are differentiable of class Cr.
Immersed submanifolds
An immersed submanifold of a manifold M is the imageS of an immersion map f: N → M; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective – it can have self-intersections. More narrowly, one can require that the map f: N → M be an injection, in which we call it an injective immersion, and define an immersed submanifold to be the image subset S together with a topology and differential structure such that S is a manifold and the inclusion f is a diffeomorphism: 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 f : N → M the image ofN in M can be uniquely given the structure of an immersed submanifold so that f : N → f 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 relative topology inherited from M. In general, it will be finer than the subspace topology. Immersed submanifolds occur in the theory ofLie groups where Lie subgroups are naturally immersed submanifolds.
Embedded submanifolds
An embedded 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 embeddingf : N → M of a manifold N in M the image f 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 letk be an integer such that 0 ≤ k ≤ n. A k-dimensional embedded submanifold of M is a subset S ⊂ M such that for every point p ∈ Sthere exists a chart containing p such that φ is the intersection of a k-dimensional plane with φ. The pairs form an atlas for the differential structure on S. 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 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 Cr submanifolds with r = 0. 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 linear subspace 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 i : S → M 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. If i is closed then S is called a closed embedded submanifold of M. Closed embedded submanifolds form the nicest class of submanifolds.
