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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2011
    • (edited Feb 2nd 2011)

    I am wondering about the following question, which possibly just demonstrates my ignorance:

    let 𝔤\mathfrak{g} be a Lie algebra and { i} i\{\langle -\rangle_i\}_i a set of indecomposable invariant polynomials. Let AA be a 𝔤\mathfrak{g}-valued differential form on an open ball/Cartesian space UU. Write F A i\langle F_A \rangle_i for its curvature characteristic forms, induced by the chose invariant polynomials.

    Then let {C i} i\{C_i\}_i be any extension of these curvature characteristic forms to closed smooth forms on the cylinder U×[0,1]U \times [0,1], hence a “smooth 1-parameter flow” of the curvature characteristic forms.

    Under which conditions can we find a flow of AA that follos this flow of curvature characteristic forms?

    In other words, under which conditions can we find an extension A^\hat A of AA to U×[0,1]U \times [0,1] such that the C i=F A^ iC_i = \langle F_{\hat A}\rangle_i?

    This is equivalently solving a system of (in general highly nonlinear) first order differential equations

    { σF A^=J i} \{ \partial_\sigma \langle F_{\hat A}\rangle = J_i \}

    for the A^\hat A given boundary conditions at σ=0\sigma = 0, subject to what should serve as an integrability conditon on the J i=d U(ι σC^ i)J_i = d_U (\iota_{\partial_\sigma} \hat C_i).