Theorems · Theorem · global analysis

VectorField.pullback_lieBracket

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {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],
  minSmoothness 𝕜 2 ≤ n →
    ∀ {f : E → F} {V W : F → F} {x : E},
      ContDiffAt 𝕜 n 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. One could also give a version avoiding completeness but requiring that f is a local diffeomorphism.

Defined in

Mathlib.Analysis.Calculus.VectorField

Cited by

0 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace