Theorems · Theorem · global analysis

VectorField.leibniz_identity_mlieBracketWithin_apply

∀ {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 3) M]
  [CompleteSpace E] {U V W : (x : M) → TangentSpace I x} {s : Set M} {x : M},
  UniqueMDiff[s] →
    x ∈ closure (interior s) →
      x ∈ s →
        ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x => ⟨x, U x⟩) s x →
          ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x => ⟨x, V x⟩) s x →
            ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x => ⟨x, W x⟩) s x →
              VectorField.mlieBracketWithin I U (VectorField.mlieBracketWithin I V W s) s x =
                VectorField.mlieBracketWithin I (VectorField.mlieBracketWithin I U V s) W s x +
                  VectorField.mlieBracketWithin I V (VectorField.mlieBracketWithin I U W s) s x

The Lie bracket of vector fields in manifolds satisfies the Leibniz identity [U, [V, W]] = [[U, V], W] + [V, [U, W]] (also called Jacobi identity).

Defined in

Mathlib.Geometry.Manifold.VectorField.LieBracket

Cited by

1 results in Mathlib

Foundations

Depth 230 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceIsManifoldCompleteSpace