Theorems · Theorem · global analysis

VectorField.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] {s : Set M} {x₀ : M} {V W : (x : M) → TangentSpace I x},
  VectorField.mlieBracketWithin I V W s x₀ =
    (mfderiv% ↑(extChartAt I x₀) x₀).inverse
      (VectorField.lieBracketWithin 𝕜
        (VectorField.mpullbackWithin (modelWithCornersSelf 𝕜 E) I (↑(extChartAt I x₀).symm) V (Set.range ↑I))
        (VectorField.mpullbackWithin (modelWithCornersSelf 𝕜 E) I (↑(extChartAt I x₀).symm) W (Set.range ↑I))
        (↑(extChartAt I x₀).symm ⁻¹' s ∩ Set.range ↑I) (↑(extChartAt I x₀) x₀))

Defined in

Mathlib.Geometry.Manifold.VectorField.LieBracket

Cited by

6 results in Mathlib

Foundations

Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpace