Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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 $J_i = d_U (\iota_{\partial_\sigma} \hat C_i)$.