{title2=$GL(n,\mathbb C)/U(n)\cong H(n)$}

Let $H(n)$ be the real <vector space> of <Hermitian matrices>. The <polar decomposition of an invertible complex matrix> gives a diffeomorphism
$$
H(n)\times U(n)\longrightarrow GL(n,\mathbb C),
\qquad (B,u)\longmapsto e^Bu.
$$
Its inverse sends $g$ to
$$
B=\frac12\log(gg^*),
\qquad u=e^{-B}g.
$$
Right 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>.


 General linear group modulo the unitary group

ID: general-linear-group-modulo-the-unitary-group