Theorems · Theorem · global analysis

contMDiffOn_iff_localFrameCoeff

∀ {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 : FiberBundle F V] [inst_13 : VectorBundle 𝕜 F V]
  [inst_14 : ContMDiffVectorBundle 1 F V I] {e : Bundle.Trivialization F Bundle.TotalSpace.proj}
  [inst_15 : MemTrivializationAtlas e] {ι : Type u_7} (b : Module.Basis ι 𝕜 F) {s : (x : M) → V x} {t : Set M}
  {k : WithTop ℕ∞} [FiniteDimensional 𝕜 F] [CompleteSpace 𝕜] [ContMDiffVectorBundle k F V I],
  IsOpen t →
    t ⊆ e.baseSet →
      (ContMDiffOn I (I.prod (modelWithCornersSelf 𝕜 F)) k (fun x => ⟨x, s x⟩) t ↔
        ∀ (i : ι),
          ContMDiffOn I (modelWithCornersSelf 𝕜 𝕜) k
            ((LinearMap.piApply (Bundle.Trivialization.localFrameCoeff I e b i)) s) t)

A section s of V is C^k on t ⊆ e.baseSet iff each of its coefficients (LinearMap.piApply (b.localFrameCoeff e i) s) in a local frame near x is

Defined in

Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Cited by

2 results in Mathlib

Foundations

Depth 225 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceFiberBundleVectorBundleContMDiffVectorBundleMemTrivializationAtlasFiniteDimensionalCompleteSpaceContMDiffVectorBundle