Theorems · Theorem · algebraic topology

Bundle.Trivialization.contMDiffWithinAt_section

∀ {n : WithTop ℕ∞} {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [inst : NontriviallyNormedField 𝕜]
  {EB : Type u_7} [inst_1 : NormedAddCommGroup EB] [inst_2 : NormedSpace 𝕜 EB] {HB : Type u_8}
  [inst_3 : TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [inst_4 : TopologicalSpace B]
  [inst_5 : ChartedSpace HB B] [inst_6 : (x : B) → AddCommMonoid (E x)] [inst_7 : (x : B) → Module 𝕜 (E x)]
  [inst_8 : NormedAddCommGroup F] [inst_9 : NormedSpace 𝕜 F] [inst_10 : TopologicalSpace (Bundle.TotalSpace F E)]
  [inst_11 : (x : B) → TopologicalSpace (E x)] [inst_12 : FiberBundle F E] [inst_13 : VectorBundle 𝕜 F E]
  [ContMDiffVectorBundle n F E IB] {s : (x : B) → E x} (a : Set B) {x₀ : B}
  {e : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e],
  x₀ ∈ e.baseSet →
    (ContMDiffWithinAt IB (IB.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, s x⟩) a x₀ ↔
      ContMDiffWithinAt IB (modelWithCornersSelf 𝕜 F) n (fun x => (↑e ⟨x, s x⟩).2) a x₀)

Smoothness of a C^n section at x₀ within a set a can be determined using any trivialisation whose baseSet contains x₀.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.Basic

Cited by

2 results in Mathlib

Foundations

Depth 213 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceAddCommMonoidModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceFiberBundleVectorBundleContMDiffVectorBundleMemTrivializationAtlas