This is a cross-post from math.SE

Let $E \to M$ be a complex vector bundle with $P$ the associated $GL(n,\mathbb C)$ frame bundle. The group of gauge transformations is the space of sections of $P\times_{Ad} GL(n,\mathbb C)$ and the space of hermitian metrics is the space of sections of $P\times_{GL(n,\mathbb C)} GL(n,\mathbb C)/U(n)$.

What is a clean way to show that the group of gauge transformations acts transitively on the space of hermitian metrics? That is, suppose $h_1, h_2$ are hermitian metrics, which are identifiable with $GL(n,\mathbb C)$-equivariant maps $P \to GL(n,\mathbb C)/U(n)$, then we need to find an equivariant map $\phi : P \to GL(n,\mathbb C)$ such that $\phi h_1 = h_2$. I could see how to get this if we know that every equivariant map $P\to GL(n,\mathbb C)/U(n)$ lifts to an equivariant map $P\to GL(n,\mathbb C)$ but I'm not sure if that's true. Maybe the fact that $GL(n,\mathbb C)/U(n)$ is contractible comes into play.

This is a cross-post from math.SE

Let $E \to M$ be a complex vector bundle with $P$ the associated $GL(n,\mathbb C)$ frame bundle. The group of gauge transformations is the space of sections of $P\times_{Ad} GL(n,\mathbb C)$ and the space of hermitian metrics is the space of sections of $P\times_{GL(n,\mathbb C)} GL(n,\mathbb C)/U(n)$.

What is a clean way to show that the group of gauge transformations acts transitively on the space of hermitian metrics? That is, suppose $h_1, h_2$ are hermitian metrics, which are identifiable with $GL(n,\mathbb C)$-equivariant maps $P \to GL(n,\mathbb C)/U(n)$, then we need to find an equivariant map $\phi : P \to GL(n,\mathbb C)$ such that $\phi h_1 = h_2$. I could see how to get this if we know that every equivariant map $P\to GL(n,\mathbb C)/U(n)$ lifts to an equivariant map $P\to GL(n,\mathbb C)$ but I'm not sure if that's true. Maybe the fact that $GL(n,\mathbb C)/U(n)$ is contractible comes into play.