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

  1. May 7, 2012 #1

    SVD

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    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.
     
    Last edited: May 7, 2012
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