Theorems · Theorem · global analysis

VectorField.mpullback_mlieBracketWithin

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {H : Type u_2} [inst_1 : TopologicalSpace H] {E : Type u_3}
  [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {H' : Type u_5} [inst_6 : TopologicalSpace H']
  {E' : Type u_6} [inst_7 : NormedAddCommGroup E'] [inst_8 : NormedSpace 𝕜 E'] {I' : ModelWithCorners 𝕜 E' H'}
  {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] [IsManifold I (minSmoothness 𝕜 2) M]
  [inst_12 : IsManifold I' (minSmoothness 𝕜 2) M'] [CompleteSpace E] {n : WithTop ℕ∞} {f : M → M'}
  {V W : (x : M') → TangentSpace I' x} {x₀ : M} {s : Set M} {t : Set M'},
  MDiffAt[t] (T% V) (f x₀) →
    MDiffAt[t] (T% W) (f x₀) →
      UniqueMDiff[s] →
        ContMDiffAt I I' n f x₀ →
          x₀ ∈ s →
            minSmoothness 𝕜 2 ≤ n →
              f ⁻¹' t ∈ nhdsWithin x₀ s →
                VectorField.mpullback I I' f (VectorField.mlieBracketWithin I' V W t) x₀ =
                  VectorField.mlieBracketWithin I (VectorField.mpullback I I' f V) (VectorField.mpullback I I' f W) s x₀

The pullback commutes with the Lie bracket of vector fields on manifolds.

Defined in

Mathlib.Geometry.Manifold.VectorField.LieBracket

Cited by

3 results in Mathlib

Foundations

Depth 228 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifoldCompleteSpace