Immersed manifold

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An immersed manifold or immersed submanifold is an object from the mathematical sub-area of differential topology . This object is more rarely called an immersed manifold , in English one usually speaks of an immersed submanifold .

If one has a differentiable mapping between two manifolds , the image is generally not a submanifold of . However, if the derivative of is injective, is a manifold, but it does not have to be a (embedded) submanifold of . This object is called the immersed manifold.

definition

Let and be differentiable manifolds . Then there is an immersed manifold of the image of immersion . The topology on must be chosen so that is continuous. Often it is still required that the immersion must be injective.

As a set is a subset of , however it is generally not a submanifold of . This means that the topology of here also does not correspond to the subspace topology and, in particular, the differentiable structures of and are not compatible. However, if there is a differentiable embedding , then there is actually a submanifold.

Differentiation to submanifold

There are two reasons why the immersed manifold need not be a submanifold:

  • The immersion is not injective , the immersion cuts itself (see Figure 1)
  • Even if the immersion is injective, the mapping may not be homeomorphism , since the open-ended image can get arbitrarily close to interior points such that the topology of does not match that of. (see Fig. 2) This effect can only occur for non- compact manifolds; for compact manifolds , an injective immersion is always an embedding.
Fig. 1: Real number line immersively mapped into the plane with self-sections
Fig. 2: Open interval mapped injectively and immersively, so that the open ends are mapped onto the ends marked with arrows

example

  • The curve defined by is an injective immersion. Hence their image is an immersed manifold.
  • A Lie group is both a group in the sense of algebra and also a smooth manifold , whereby the two structures are compatible with each other. A Lie subgroup is a subgroup of the Lie group, which again has the structure of a smooth manifold that is compatible with the group structure. This Lie subgroup is generally not a submanifold, but an immersed (sub) manifold, where the immersion is injective. A concrete example is a curve of irrational increase in the torus. This is a subgroup and an immersed submanifold, but not embedded: its image lies close to the torus.

literature

Individual evidence

  1. ^ Stefan Hildebrandt : Analysis. Volume 2. Springer, Berlin et al. 2003, ISBN 3-540-43970-6 .
  2. The correct derivation from Latin is actually "immersed manifold", but in German the "immersed manifold" derived from the English "immersed manifold" has recently become a more common variant.
  3. John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 , p. 15.