Theorems · Theorem · differential geometry

IsCovariantDerivativeOn.torsion_apply_eq_extend

∀ {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 2 M] [inst_7 : CompleteSpace E]
  {cov : ((x : M) → TangentSpace I x) → (x : M) → TangentSpace I x →L[𝕜] TangentSpace I x} [inst_8 : CompleteSpace 𝕜]
  [inst_9 : FiniteDimensional 𝕜 E] (hcov : IsCovariantDerivativeOn E cov) {x : M} (X₀ Y₀ : TangentSpace I x),
  ((hcov.torsion x) X₀) Y₀ =
    (cov (FiberBundle.extend E Y₀) x) X₀ - (cov (FiberBundle.extend E X₀) x) Y₀ -
      VectorField.mlieBracket I (FiberBundle.extend E X₀) (FiberBundle.extend E Y₀) x

Defined in

Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion

Cited by

2 results in Mathlib

Foundations

Depth 230 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldCompleteSpaceCompleteSpaceFiniteDimensional