Does anyone know a citeable reference which works out the properties (geodesics, geodesic distance, ect) of the Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$, $m>n$, with the Hilbert-Schmidt inner product on the tangent space? (An isometry is a linear map A so that $A^\*A=1$.)


Even knowing the proper name of this manifold would be useful, so that I could search for it.


(This manifold is related to some work in quantum detection theory. I tried mathstackexchange, but didn't get a correct answer.)



Reference request: Riemannian manifold of  linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$
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. 


Riemannian geometry on homogeneous manifolds is governed by the Nomizu operator.
See page 364-367 of [here](http://www.mat.univie.ac.at/~michor/dgbook.pdf).


Explicit geodesics for Grassmannians, even a formula for geodesic distance, in the form of horizobtal geodesics of the Stiefel manifold, are described in the following paper:


* Y.~A. Neretin: On Jordan angles and the triangle inequality in
Grassmann manifold}, Geometriae Dedicata, 86 (2001).


Geodesics on an infinite dimensional Stiefel manifold of isometries of $\mathbb R^2$ into a Hilbert space were used for shape space ananlysis in the paper


* Laurent Younes, Peter W. Michor, Jayant Shah, David Mumford: A Metric on Shape Space with Explicit Geodesics. Rend. Lincei Mat. Appl. 9 (2008) 25-57. arXiv:0706.4299 [(pdf)](http://www.mat.univie.ac.at/~michor/H1inf.pdf)


