How do I get the 1st fundamental form on Grassmann Manifold

[SVD]

Consider G(n,m), the set of all n-dimensional subsapce in ℝ^n+m.
We define the principal angles between two subspaces recusively by the usual formula.
When I see "Differential Geometry of Grassmann Manifolds by Wong",
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC335549/pdf/pnas00676-0108.pdf
I have no idea how to derive the 1st fundamental form given in theorem 4. In particular, I want to know how to derive
this formula when the underlying scalar field are real.

Since every n-plane can be expressed as a system of m linear homogeneous equation in m+n variables, we
can describe this n-plane by a m*(m+n) matrix with rank m. W.L.O.G we can assume the last m column are linear independent.
Divide this m*(m+n)matrix into m*n and m*m matrix in the sense A(x1,...xn)+B(xn+1,...xn+m)=O. Since B is invertible, we have
(xn+1,...xn+m)=B^-1*-A(x1,...xn), and this defines a local chart for the Grassmann manifold. Other kinds of local chart arises if
the m linear independent columns does not lie on the last m columns.
Denote B^-1*-A=Z, an m*n matrix defines a local chart for the Grassmann manifold. Theorem 4 states that the 1st fumdamental
form in this local chart can be expressed as ds^2=Tr[(I+ZZ^t)^-1dZ(I+Z^tZ)^-1dZ^t] if the underlying scalar field are real.
Since this paper doesn't provide any proof, it just states theorem and I want to know how to derive the above 1st fumdamental
form in Z local chart and the result as this paper states will make the distance between two n planes equal sum of squares of their principal angle.

 

[jvicens]

Thank you for your interest in the paper "Differential Geometry of Grassmann Manifolds" by Wong. The derivation of the 1st fundamental form given in theorem 4 can be found in the paper "The Geometry of Grassmann Manifolds" by Helgason. In this paper, Helgason proves the general result for complex scalar fields, but the same argument can be applied for real scalar fields as well.

The key idea behind the derivation is to use the fact that the Grassmann manifold G(n,m) can be identified with the quotient space of the special linear group SL(n+m,R) by the subgroup K=SL(n,R)×SL(m,R). This identification is known as the Iwasawa decomposition and it allows us to use the group structure of SL(n+m,R) to study the geometry of G(n,m).

In particular, the 1st fundamental form can be derived by considering the action of the subgroup K on the tangent space at the identity element of SL(n+m,R). This tangent space can be identified with the space of m*(m+n) matrices of the form Z=B^-1*-A, where B is an m*m invertible matrix and A is an m*n matrix. This is exactly the local chart you described in your post.

The action of K on this tangent space induces a metric on the Grassmann manifold, which turns out to be the 1st fundamental form given in theorem 4. The details of this construction can be found in Helgason's paper.

I hope this helps to clarify the derivation of the 1st fundamental form in theorem 4. If you have any further questions, please do not hesitate to ask. Thank you for your interest in this topic.