Why are Lie Subgroups Defined as Immersed Submanifolds?

[center o bass]

A Lie Subgroup is defined as follows:

A Lie subgroup of a Lie group G is (i) an abstract subgroup H that is (ii) an immersed submanifold via the inclusion map such that (iii) the group operations on H are smooth.

While a Lie Group is defined as a group with smooth multiplication and inversion maps, that is also a manifold. One would think that a Lie subgroup would be defined so that it is also a Lie group in it's own right. However, immersed submanifolds are not embedded submanfolds, and thus not in general manifolds in their own right.

What is the reason for this definition?

 

[micromass]

But an immersed submanifold is a general manifold in its own right. What makes you say it's not a manifold?

The reason for the definition is because we want images of homomorphisms to be subgroups. Thus if $G$ is a Lie group and if $f:G\rightarrow H$ is a smooth group homomorphism, then we want $f(G)$ to be a subgroup. But if we restrict the notion of subgroup to 'embedded', then this is false. So we need them to be 'immersed' only.

 

[center o bass]

Let me quote Lee Tu - An introduction to Manifolds page 122

" If the underlying set of an immersed submanifold is given the
subspace topology, then the resulting space need not be a manifold at all!"

 

[micromass]

Let me quote Lee Tu - An introduction to Manifolds page 122

" If the underlying set of an immersed submanifold is given the
subspace topology, then the resulting space need not be a manifold at all!"

True. But you usually don't give it the subspace topology! So you should make a distinction between "the underlying set with the subspace topology" and the immersed submanifold.

If $M$ is a manifold, then an immersed submanifold is a manifold $N$ with an injective immersion $i:N\rightarrow M$. So the immersed submanifold $N$ certainly is a manifold in its own sense, but $i$ might not be an embedding, thus the topology of $N$ and the subspace topology of $i(N)$ might have nothing to do with each other.

I don't think you should see an immersed submanifold as a subset of the manifold. A better way to see it is as a manifold such that an immersion to the big manifold exists. This is called the categorical view of "substructures": that is, to see a substructure as a mapping into the space instead of as an actual subset. This view becomes very important in more advanced mathematics, such as algebraic geometry.