I am currently reading the book "Lectures on Kähler Geometry" by Andrei Moroianu and I try to understand the following theorem :
The proof of this theorem is given in the book, but unfortunately the author does not explain why $G = Gl_k(\mathbb{R})$ and the identity $\rho$ are inverses to each other. I would be grateful if someone can explain this to me. Note that $Gl_k(\mathbb{R})$ is the set of frames at $k$.
Thanks for your help.