General linear group modulo the unitary group ($GL(n,\mathbb{C})/U(n)\cong H(n)$)

Let $H(n)$ be the real vector space of Hermitian matrices. The polar decomposition of an invertible complex matrix gives a diffeomorphismIts inverse sends $g$ toRight multiplication by $U(n)$ changes only $u$, so the orbit space is smoothly $H(n)$ and the quotient map is a globally trivial principal bundle.