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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.)

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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:

  • 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)
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  • $\begingroup$ Yeah, that too! :-) Although I think that if this is coming from "quantum detection theory" he could do worse than to look at more general partial isometries as well. $\endgroup$
    – alvarezpaiva
    Commented May 17, 2013 at 8:12
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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.

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  • $\begingroup$ Not partial isometries, $m>n$. $\endgroup$
    – J Tyson
    Commented May 17, 2013 at 7:44
  • $\begingroup$ @Tysin, it is clear that your question is not sufficiently transparent as to what exactly it is asking for —maybe elaborating more on what you are looking for will make it easier for you to find it. Your comment here is also rather opaque :-) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented May 17, 2013 at 7:49
  • $\begingroup$ I added the precise definition of isometry to the question. (Is something else unclear?) $\endgroup$
    – J Tyson
    Commented May 17, 2013 at 7:55
  • $\begingroup$ I don't think it matters that $m > n$. The kernel of the maps has dimension zero and the map is an isometry on the space perpendicular to the kernel. From another viewpoint: if $A$ is your map $A = AA^*A$, which characterizes partial isometries. $\endgroup$
    – alvarezpaiva
    Commented May 17, 2013 at 8:01
  • $\begingroup$ Thanks! Peter Michor just pointed out the technical term, and your reference is very close. :) $\endgroup$
    – J Tyson
    Commented May 17, 2013 at 8:10

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