structure of Riemannian manifold of isometries from C^n to C^m

Question

Does anyone know a reference which gives the properties (geodesics, geodesic distance, etc) of the Riemannian manifold of isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$, $m>n$, which map zero to zero? The metric on the tangent space is inherited from the Frobenius inner product on the set of all matrices.

I'm preferably looking for a reference that I can cite in a compact publication. (This manifold is relevant to quantum detection theory.)

(Maybe this manifold has a name that someone could provide which would allow me to simply Google it.)

Answer 1

Quoting the answer by Peter Michor:

They are called Stiefel manifolds, and are principal $U(n)$-bundles over Grassmann manifolds. They are homogeneous Riemannian manifolds, whereas the Grassmannian are symmetric spaces.

Turns out they came up a few times on Math.SE.