page_title: Submanifold
section_title: Immersed submanifolds
hierarchical_section_title: Submanifold / Formal definition / Immersed submanifolds
caption_attribution_description: immersed submanifold 2
context_page_description: In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.
context_section_description: An immersed submanifold of a manifold M is the image S 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 (one-to-one) – it can have self-intersections.
  More narrowly, one can require that the map f: N → M 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 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 of N in M can be uniquely given the structure of an immersed submanifold so that f : N → f(N) 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 (i.e. have more open sets).
  Immersed submanifolds occur in the theory of Lie groups where Lie subgroups are naturally immersed submanifolds.
caption: Immersed submanifold open interval with interval ends mapped to arrow marked ends
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