Submanifold#Immersed submanifolds

An ever- ized manifold or always ized submanifold is an object from the mathematical branch of differential topology. Rare this object is also called always lusted manifold, in English we speak mostly of a immersed submanifold.

If one has a differentiable map between two manifolds, so the picture is not generally submanifold of. However, if the derivative of is injective, is a manifold, but no (embedded ) submanifold of his needs. This object is called always ized manifold.

Definition

Let and be differentiable manifolds. Then is an increasingly ized manifold of the image of the immersion. The topology should be chosen so that is continuous. It is often still required that the immersion must be injective.

As set is a subset of, but it is generally not submanifold of. That is, the topology of not here corresponds to the subspace topology and in particular, the structures of differentiable and not compatible. However, if a differentiable embedding, so is actually a submanifold.

Distinction to the submanifold

There are two reasons why the ever- ized manifold need not be a submanifold:

• The immersion is not injective immersion intersects itself (see Figure 1)
• Even if the immersion is injective, it may be that the figure is not a homeomorphism, since the image of open ends interior points can come from as close as desired, so that the topology of not that of matches. ( Fig. 2)

Example

• The curve is defined by: is an injective immersion. Therefore, their image is a still ized manifold.
• A Lie group is a group both in terms of the algebra that is also a smooth manifold, the two structures are compatible with each other. A Lie subgroup is a subgroup of the Lie group, which also again carries the structure of a smooth manifold that is compatible with the group structure. This Lie subgroup is generally not a submanifold but always ized (sub ) manifold, where the immersion is injective.