Theorems · Theorem · differential geometry

IsCovariantDerivativeOn.torsion_apply

∀ {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} {X : (x : M) → TangentSpace I x},
  MDiffAt (T% X) x →
    ∀ {Y : (x : M) → TangentSpace I x},
      MDiffAt (T% Y) x →
        ((hcov.torsion x) (X x)) (Y x) = (cov Y x) (X x) - (cov X x) (Y x) - VectorField.mlieBracket I X Y x

Defined in

Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion

Cited by

0 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldCompleteSpaceCompleteSpaceFiniteDimensional