Proof of $G\rightarrow G/H$ is a Principal H bundle

Question

Let $G$ be a Lie group and let $H$ be a closed subgroup (not necessarily normal). Then $G$ is a principal $H$-bundle over the (left) coset space $G/H$.

I could proof that the fibers are all isomorphic to H

Let $\pi: G \rightarrow G/H$ be the canonical map

$\pi^{-1}({x+H})=\{g\in G | g+H=x+H\}=\{g\in G | g^{-1}.x\in H\}$

We can define a map $\pi^{-1}({x+H})\rightarrow H$ via sending $g$ to $g^{-1}.x$.

It can be easily checked that the map is a homeomorphism.

Therefore, fibers are homeomorphic to $H$.

But I am not able to see why $H$ is the Structure group.

Lets take two distinct fibers $\pi^{-1}({x+H})$ and $\pi^{-1}(y+H)$ (i.e., $x^{-1}.y\notin H)$

Multiplying by $y.x^{-1} $ will send element of $\pi^{-1}({x+H})$ to element of $\pi^{-1}(y+H)$. But then $ y.x^{-1} $ doesn't belong to $H$.

For G to be a principal $H $ bundle over $G/H$ the structure group has to be $H$ and $H$ acting on fiber $H$ be left multiplication.

Answer 1

Part of the confusion here is that in any principal $H$-bundle $\pi\colon P\to M$, there are two different $H$-actions:

1. There is a free right action of $H$ on the total space of the bundle, whose orbits are the fibers of $\pi$.
2. There is also the left action of $H$ on itself by left multiplication, which shows up in the transition functions for the bundle. To say that $P$ is a principal $H$-bundle means that there is a covering of $M$ by open sets $\{U_\alpha\}$ together with local trivializations $\Phi_\alpha\colon \pi^{-1}(U_\alpha) \to U_\alpha \times H$, each of which is a diffeomorphism, and such that whenever $U_\alpha\cap U_\beta\ne\emptyset$, there is a smooth transition function $\tau_{\alpha\beta}\colon U_\alpha\cap U_\beta\to H$ such that for all $(x,h)\in U_\alpha\cap U_\beta\times H$, $$ \Phi_\beta\circ \Phi_\alpha^{-1}(x,h) = (x, \tau_{\alpha\beta}(x)h), $$ where $\tau_{\alpha\beta}(x)$ acts on $H$ by left multiplication. This is what it means to say the structure group of $P$ is the group $H$ acting on itself by left multiplication.

In the special case in which $P=G$ and $H$ is a closed subgroup of $G$, the right action of $H$ on $G$ is just ordinary right multiplication, and $M = G/H$ is a smooth quotient manifold. The local trivializations are obtained as follows: Given any $x_0\in G/H$, because $\pi\colon G\to G/H$ is a smooth surjective submersion, it has a smooth local section $\sigma\colon U\to G$ (i.e., smooth map such that $\pi\circ\sigma=\operatorname{id}_U$) defined on some neighborhood $U$ of $x_0$. If we define $\alpha\colon U\times H\to \pi^{-1}(U)\subset G$ by $\alpha(x,h) = \sigma(x)g$, then it is straightforward to show that $\alpha$ is a diffeomorphism, so we can define a local trivialization $\Phi\colon \pi^{-1}(U)\to U\times H$ by $\Phi = \alpha^{-1}$.

Given any other such local section $\sigma'\colon U'\to G$ and corresponding maps $\alpha'$ and $\Phi'$, a computation shows that $\Phi'\circ\Phi^{-1}(x,h) = (x,\sigma(x)\sigma'(x)^{-1}h)$. Thus each transition function is given by $\tau(x) = \sigma(x)\sigma'(x)^{-1}$, acting on $H$ by left multiplication.