How to understand immersed submanifold?

Question

I read this from Loring Tu's book:

I am so confused what "the topology and differentiable structure inherited from $f$" really means. This one doesn't stuck me when I first went here. But in the following chapter of Lie groups, I run through this again:

I wonder if there is a given smooth structure on $H$ itself. I can hardly understand "an immersed submanifold via the inclusion map". If here we can define "immersion", we must first give $H$ a smooth structure. I am so confused.

Any help will be appreciated.

Answer 1

Smoothness on $H$ is guaranteed via the definition of a submanifold. This canonically implies that the inclusion is smooth. In Differentiable Manifolds and Lie Groups by Warner, the following is claimed:

Let $M$ be a differentiable manifold and $A$ be a subset of $M$. Fix a topology on $A$. Then there is at most one differentiable structure on $A$ such that $(A,i)$ is a submanifold of $M$, where $i$ is the inclusion map.

How this should be interpreted (in my humble opinion) is given a topology on $A$, there is one and only one differentiable structure on $A$ that makes $(A, i)$ a submanifold of $M$.

Answer 2

I will first try to explain what he means with "topology and differentiable structure inherited by $f$".

We start with manifolds $M,N$ and a injective map $f:N\to M$. We define a topology on $f(N)$ in the following way: $U\subset N$ is open if and only if $f^{-1}(U)\subset N$ is open. This is the topology inhereted by $f$.

Now take a chart $\varphi':V\subset N\to \mathbb{R}^k$ around a point $x \in N$. This defines a chart $\varphi=\varphi'\circ f^{-1}:f(V)\to \mathbb{R}^k$. We have to check that the transition between two such maps is a smooth map from $\mathbb{R}^k\to \mathbb{R}^k$ but this is easy, let $\phi,\varphi$ be two such charts than you have:

\begin{eqnarray} \varphi\circ\phi^{-1}&=&\varphi'\circ f^{-1}\circ f \circ \phi'^{-1}\\ &=&\varphi'\circ \phi'^{-1} \end{eqnarray}

which is smooth by assumption that $N$ is a smooth manifold. This is the smooth structure inhereted by $f$. Notice that this makes $f(N)$ into a smooth manifold but the topology on $f(N)$ may not agree with the subspace topology. Moreso $f(N)$ may not be a manifold in the subspace topology.

Now in your example a Lie subgroup is a subgroup $H \subset G$ which is Lie group s.t. the inclusion map is a injective immersion. Notice that here immersion usualy also means that the derivative of the inclusion is injective.

Let me know if you have any question.