Theorems · Theorem · differential geometry

ContMDiffCovariantDerivativeOn.contMDiff

∀ {𝕜 : Type u_1} {inst : NontriviallyNormedField 𝕜} {E : Type u_2} {inst_1 : NormedAddCommGroup E}
  {inst_2 : NormedSpace 𝕜 E} {H : Type u_3} {inst_3 : TopologicalSpace H} {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  {inst_4 : TopologicalSpace M} {inst_5 : ChartedSpace H M} {F : Type u_5} {inst_6 : NormedAddCommGroup F}
  {inst_7 : NormedSpace 𝕜 F} {V : M → Type u_6} {inst_8 : TopologicalSpace (Bundle.TotalSpace F V)}
  {inst_9 : (x : M) → AddCommGroup (V x)} {inst_10 : (x : M) → Module 𝕜 (V x)}
  {inst_11 : (x : M) → TopologicalSpace (V x)} {inst_12 : ∀ (x : M), IsTopologicalAddGroup (V x)}
  {inst_13 : ∀ (x : M), ContinuousSMul 𝕜 (V x)} {inst_14 : FiberBundle F V} {inst_15 : IsManifold I 1 M}
  {inst_16 : VectorBundle 𝕜 F V} {k : WithTop ℕ∞} {cov : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x}
  {u : Set M} [self : ContMDiffCovariantDerivativeOn F k cov u] {σ : (x : M) → V x},
  ContMDiffOn I (I.prod (modelWithCornersSelf 𝕜 F)) (k + 1) (fun x => ⟨x, σ x⟩) u →
    ContMDiffOn I (I.prod (modelWithCornersSelf 𝕜 (E →L[𝕜] F))) k (fun x => ⟨x, cov σ x⟩) u

Defined in

Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic

Cited by

2 results in Mathlib

Foundations

Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ContMDiffCovariantDerivativeOn