Theorems · Theorem · global analysis

ContMDiffWithinAt.mlieBracketWithin_vectorField

∀ {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] [inst_6 : IsManifold I (minSmoothness 𝕜 2) M]
  [CompleteSpace E] {n : WithTop ℕ∞} [IsManifold I (n + 1) M] {m : WithTop ℕ∞} {U V : (x : M) → TangentSpace I x}
  {s : Set M} {x : M},
  ContMDiffWithinAt I I.tangent n (fun x => ⟨x, U x⟩) s x →
    ContMDiffWithinAt I I.tangent n (fun x => ⟨x, V x⟩) s x →
      UniqueMDiff[s] →
        x ∈ s →
          minSmoothness 𝕜 (m + 1) ≤ n →
            ContMDiffWithinAt I I.tangent m (fun x₀ => ⟨x₀, VectorField.mlieBracketWithin I U V s x₀⟩) s x

If two vector fields are C^n with n ≥ m + 1, then their Lie bracket is C^m.

Defined in

Mathlib.Geometry.Manifold.VectorField.LieBracket

Cited by

3 results in Mathlib

Foundations

Depth 229 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceIsManifoldCompleteSpaceIsManifold