Theorems · Theorem · global analysis

TangentBundle.symmL_trivializationAt

∀ {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] [inst_6 : IsManifold I 1 M] {x₀ x : M},
  x ∈ (chartAt H x₀).source →
    Bundle.Trivialization.symmL 𝕜 (trivializationAt E (TangentSpace I) x₀) x =
      mfderiv[Set.range ↑I] ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x)

The inverse trivialization of the tangent bundle at a point is the manifold derivative of the inverse of the extended chart. Use with care as this abuses the defeq TangentSpace 𝓘(𝕜, E) y = E for y : E.

Defined in

Mathlib.Geometry.Manifold.MFDeriv.Atlas

Cited by

1 results in Mathlib

Foundations

Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifold