Theorems · Theorem · algebraic topology

ContMDiffOn.smul_section_of_tsupport

∀ {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] {n : WithTop ℕ∞} {V : M → Type u_6} [inst_8 : TopologicalSpace (Bundle.TotalSpace F V)]
  [inst_9 : (x : M) → TopologicalSpace (V x)] [inst_10 : FiberBundle F V] [inst_11 : (x : M) → AddCommGroup (V x)]
  [inst_12 : (x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] {u : Set M} {s : (x : M) → V x} {ψ : M → 𝕜},
  ContMDiffOn I (modelWithCornersSelf 𝕜 𝕜) n ψ u →
    IsOpen u →
      tsupport ψ ⊆ u →
        ContMDiffOn I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, s x⟩) u →
          ContMDiff I (I.prod (modelWithCornersSelf 𝕜 F)) n fun x => ⟨x, (ψ • s) x⟩

The scalar product ψ • s of a C^k function ψ : M → 𝕜 and a section s of a vector bundle V → M is C^k once s is C^k on an open set containing tsupport ψ. This is a vector bundle analogue of contMDiff_of_tsupport.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Cited by

1 results in Mathlib

Foundations

Depth 211 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceFiberBundleAddCommGroupModuleVectorBundle