Theorems · Theorem · global analysis

Bundle.Trivialization.localFrameCoeff_eq_coeff

∀ {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] {ι : Type u_7}
  (e : Bundle.Trivialization F Bundle.TotalSpace.proj) [inst_14 : MemTrivializationAtlas e] {b : Module.Basis ι 𝕜 F}
  [inst_15 : ContMDiffVectorBundle 1 F V I] {x : M} {s : (x : M) → V x},
  x ∈ e.baseSet →
    ∀ {i : ι}, (Bundle.Trivialization.localFrameCoeff I e b i x) (s x) = (b.repr (↑e ((fun x => ⟨x, s x⟩) x)).2) i

Suppose e is a compatible trivialisation around x ∈ M, and s a bundle section. Then the coefficient of s w.r.t. the local frame induced by b and e equals the coefficient of "s x read in the trivialisation e" for b i.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Cited by

3 results in Mathlib

Foundations

Depth 222 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceFiberBundleVectorBundleMemTrivializationAtlasContMDiffVectorBundle