What is the significance of immersions and embeddings in the study of manifolds?

[Cincinnatus]

My professor recently defined immersions and embeddings in class, but he didn't really make any attempt to motivate these definitions. Now a few weeks later, after we've been using these concepts in class I'm still having trouble understanding what is *natural* about these definitions, what they *really mean* is still murky... I wonder if there's anyone here who can help me motivate these definitions?

Here's the actual definitions used:
Immersion: a function f from one manifold M to another N is an immersion if it's differential df_p, mapping from a tangent space T_pM to T_f(p)N is injective.

Embedding: An immersion is an embedding if it is 1-1 and it's inverse is continuous.

 

[Hurkyl]

Compare with the notion of a submanifold.

 

[Cincinnatus]

Well Spivak's book says there are two kinds of submanifolds, immersed submanifolds (when the inclusion map is a immersion) and embedded submanifolds (when the inclusion map is an embedding). After thinking about it, embedded submanifolds seem to be in agreement with my previous notion of what a submanifold was. I'm still not really sure what an immersed submanifold is though...

 

[Hurkyl]

Immersion is the local version of embedding: each point of the source has a neighborhood U such that if we restrict the immersion to U, we get an actual embedding.

An example of an immersion that is not an embedding is the map sending a circle to a figure-8 subset of the plane.

 

[mathwonk]

an immersion is a local version of an embedding. to understand the global version, study the concept of proper mapping.

 

[Cincinnatus]

hmm... I'm don't think I see how an immersion is a local version of an embedding. To me saying a property is "local" has always meant "true in a neighborhood of each point" rather than "true for the whole set". Is this not what you mean by local?

 

[Cincinnatus]

ohh, I understand now, my professor spent today's class discussing this.

 

[mathwonk]

great!

 

[wofsy]

An immersion allows one to study the geometry of a manifold, or its physics, in terms of the larger manifold into which it is immersed. For example a smooth curve immersed in space has a velocity and an acceleration (Physics) and curvature and torsion (geometry).

It its differential were not injective then it would have points of zero velocity which really OK but if it is kinked or broken will not have any physics or smooth geometry.

The best way not to lose any information is to require that the derivative be injective. This way the calculus of the manifold is completely preserved in the larger space.

Examples:

In geometry, a classical problem is whether a manifold with a given geometry or topology can be immersed or embedded in another manifold. For instance a classical result of Hilbert says that any compact surface in 3 space must have a point of positive curvature. So a flat torus can not be embedded (or immersed) in 3 space. However, it can be embedded in the sphere in 4 space.

A Klein bottle can be immersed in 3 space but not embedded. This is because a hypersurface of Euclidean space must be orientable - which the Klein bottle is not. However the Klein bottle can be embedded in 4 space. This shows that the problem of immersion and embedding are different in an interesting way.

The hyperbolic plane can not be immersed in 3 space but if you do not require smoothness of the mapping then it can (I think).

Any manifold is naturally embedded in its tangent bundle as the set of zero tangent vectors. This embedding contains information about the topology of the manifold such as its Euler characteristic.