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.

 

[blue_raver22]

The reason for defining Lie subgroups as immersed submanifolds is to ensure that they inherit the smooth structure of the larger Lie group. This allows for a natural and consistent way to define smooth group operations on the subgroup, making it a Lie group in its own right.

Moreover, the immersion property ensures that the subgroup is locally diffeomorphic to the larger Lie group, preserving important geometric and algebraic properties. This is important in the study of Lie groups and their subgroups, as it allows for the application of powerful tools and techniques from differential geometry and Lie theory.

Additionally, defining Lie subgroups as immersed submanifolds also allows for a more general and flexible definition, as embedded submanifolds may not always exist for certain subgroups. This definition allows for a wider range of subgroups to be studied within the framework of Lie theory.

In summary, defining Lie subgroups as immersed submanifolds is a crucial aspect of the theory of Lie groups, ensuring a smooth and consistent structure for subgroups and allowing for a wider range of subgroups to be studied.