Theorems · Theorem · global analysis

VectorField.pullback_lieBracket_of_isSymmSndFDerivAt

∀ {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] [CompleteSpace E]
  {f : E → F} {V W : F → F} {x : E},
  IsSymmSndFDerivAt 𝕜 f x →
    ContDiffAt 𝕜 2 f x →
      DifferentiableAt 𝕜 V (f x) →
        DifferentiableAt 𝕜 W (f x) →
          VectorField.pullback 𝕜 f (VectorField.lieBracket 𝕜 V W) x =
            VectorField.lieBracket 𝕜 (VectorField.pullback 𝕜 f V) (VectorField.pullback 𝕜 f W) 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

1 results in Mathlib

Foundations

Depth 208 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace