Theorems · Theorem · global analysis

Bundle.Trivialization.mdifferentiableOn_section_iff

∀ {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F] [inst_3 : TopologicalSpace (Bundle.TotalSpace F E)]
  [inst_4 : (x : B) → TopologicalSpace (E x)] {EB : Type u_7} [inst_5 : NormedAddCommGroup EB]
  [inst_6 : NormedSpace 𝕜 EB] {HB : Type u_8} [inst_7 : TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB}
  [inst_8 : TopologicalSpace B] [inst_9 : ChartedSpace HB B] [inst_10 : FiberBundle F E]
  [inst_11 : (x : B) → AddCommMonoid (E x)] [inst_12 : (x : B) → Module 𝕜 (E x)] [inst_13 : VectorBundle 𝕜 F E]
  [ContMDiffVectorBundle 1 F E IB] {s : (x : B) → E x} {a : Set B} (e : Bundle.Trivialization F Bundle.TotalSpace.proj)
  [MemTrivializationAtlas e], IsOpen a → a ⊆ e.baseSet → (MDiff[a] (T% s) ↔ MDiff[a] fun x => (↑e ⟨x, s x⟩).2)

Differentiability of a section on s can be determined using any trivialisation whose baseSet contains s.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Cited by

1 results in Mathlib

Foundations

Depth 216 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceFiberBundleAddCommMonoidModuleVectorBundleContMDiffVectorBundleMemTrivializationAtlas