Theorems · Theorem · differential geometry

CovariantDerivative.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] {x : M} [inst_6 : CompleteSpace 𝕜]
  [inst_7 : CompleteSpace E] [inst_8 : FiniteDimensional 𝕜 E] [inst_9 : IsManifold I 2 M]
  (cov : CovariantDerivative I E (TangentSpace I)) {X Y : (x : M) → TangentSpace I x},
  MDiffAt (T% X) x →
    MDiffAt (T% Y) x →
      ((cov.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

1 results in Mathlib

Foundations

Depth 231 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceCompleteSpaceCompleteSpaceFiniteDimensionalIsManifold