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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.)

Jesse Madnick
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Jon Tyson
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  • I'm afraid I don't know of any relevant literature, but I suppose the proof of Wigner's theorem might yield a description of your manifold, at least as a set? In particular, given an inclusion $\mathbb{C}^n \hookrightarrow \mathbb{C}^m$, your manifold will include all isometries on $\mathbb{C}^n$, which by Wigner's theorem is the group generated by $U(n)$ together with the entry-wise complex conjugation operator $C$ on $\mathbb{C}^n$. – Branimir Ćaćić May 08 '13 at 10:54
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    A linear map $T:\mathbb C^n\to \mathbb C^m$ is an isometry if and only if its adjoint $T^*$ is an orthogonal projection. In this way the manifold of isometries can be identified with the manifold of rank $n$ projections, the Grassmannian. – 75064 May 10 '13 at 05:24
  • You are mistaken. Consider, for example, the case that $m=n$. Do you still claim that T is unitary if and only if T* is a projection???? If $m>n$ then $T^*$ does not even map the vector space to itself, so it can't be a projection. – Jon Tyson May 17 '13 at 06:07
  • I got the answer on mathoverflow. The proper name for this space is a "Stiefel manifold". – Jon Tyson May 17 '13 at 09:23

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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.

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