Theorems · Theorem · global analysis

TangentBundle.continuousLinearMapAt_trivializationAt_eq_core

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {H : Type u_4} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] [inst_6 : IsManifold I 1 M] {b₀ b : M},
  b ∈ (chartAt H b₀).source →
    Bundle.Trivialization.continuousLinearMapAt 𝕜 (trivializationAt E (TangentSpace I) b₀) b =
      (tangentBundleCore I M).coordChange (achart H b) (achart H b₀) b

The trivialization of the tangent space can be expressed in terms of the tangent bundle core. To write it as the manifold derivative of extChartAt, see TangentBundle.continuousLinearMapAt_trivializationAt. Use with care as it abuses the defeq TangentSpace I b = E.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.Tangent

Cited by

2 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifold