Transitivity of the action of the group of gauge transformations on the space of hermitian metrics

Question

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.

Answer 1

View $h_i$ as conjugate linear bundle isomorphisms $h_i:E\to E^*$. Then $\phi=h_1^{-1}\circ h_2$ satisfies $h_2(x,y)=h_1(\phi.x,y)$. Since $h_2$ is positive hermitian, the endomorphism $\phi$ is positive hermitian with respect to $h_1$, thus $\sqrt{\phi}$ exists with respect to $h_1$ and $h_2(x,y)=h_1(\sqrt{\phi}.x,\sqrt{\phi}.y)$.