Theorems · Theorem · global analysis

VectorField.pullbackWithin_lieBracketWithin_of_isSymmSndFDerivWithinAt

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E}
  [CompleteSpace E] {f : E → F} {V W : F → F} {x : E} {t : Set F},
  IsSymmSndFDerivWithinAt 𝕜 f s x →
    ContDiffWithinAt 𝕜 2 f s x →
      DifferentiableWithinAt 𝕜 V t (f x) →
        DifferentiableWithinAt 𝕜 W t (f x) →
          UniqueDiffOn 𝕜 s →
            x ∈ s →
              Set.MapsTo f s t →
                VectorField.pullbackWithin 𝕜 f (VectorField.lieBracketWithin 𝕜 V W t) s x =
                  VectorField.lieBracketWithin 𝕜 (VectorField.pullbackWithin 𝕜 f V s)
                    (VectorField.pullbackWithin 𝕜 f W s) s x

The Lie bracket commutes with taking pullbacks. This requires the function to have symmetric second derivative. Version in a complete space. One could also give a version avoiding completeness but requiring that f is a local diffeomorphism.

Defined in

Mathlib.Analysis.Calculus.VectorField

Cited by

3 results in Mathlib

Foundations

Depth 207 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace