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


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*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


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*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)
A: At first sight, it seems to me you talking about partial isometries. If so,  this paper by Andruchow and Corach (and/or its references) may help.
