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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 2nd 2011
   • (edited Feb 2nd 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am wondering about the following question, which possibly just demonstrates my ignorance: let $\mathfrak{g}$ be a Lie algebra and $\{\langle -\rangle_i\}_i$ a set of indecomposable invariant polynomials. Let $A$ be a $\mathfrak{g}$-valued differential form on an open ball/Cartesian space $U$. Write $\langle F_A \rangle_i$ for its curvature characteristic forms, induced by the chose invariant polynomials. Then let $\{C_i\}_i$ be any extension of these curvature characteristic forms to closed smooth forms on the cylinder $U \times [0,1]$, hence a "smooth 1-parameter flow" of the curvature characteristic forms. Under which conditions can we find a flow of $A$ that follos this flow of curvature characteristic forms? In other words, under which conditions can we find an extension $\hat A$ of $A$ to $U \times [0,1]$ such that the $C_i = \langle F_{\hat A}\rangle_i$? This is equivalently solving a system of (in general highly nonlinear) first order differential equations $$ \{ \partial_\sigma \langle F_{\hat A}\rangle = J_i \} $$ for the $\hat A$ given boundary conditions at $\sigma = 0$, subject to what should serve as an integrability conditon on the $J_i = d_U (\iota_{\partial_\sigma} \hat C_i)$.

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

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

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

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

   In other words, under which conditions can we find an extension $\hat A$ of $A$ to $U \times [0,1]$ such that the $C_i = \langle F_{\hat A}\rangle_i$?

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

   $$\{ \partial_\sigma \langle F_{\hat A}\rangle = J_i \}$$

   for the $\hat A$ given boundary conditions at $\sigma = 0$, subject to what should serve as an integrability conditon on the $J_i = d_U (\iota_{\partial_\sigma} \hat C_i)$.