Theorems · Theorem · differential geometry

CovariantDerivative.torsion_eq_zero_iff

∀ {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 : CompleteSpace 𝕜] [inst_7 : CompleteSpace E]
  [inst_8 : FiniteDimensional 𝕜 E] [inst_9 : IsManifold I 2 M] (cov : CovariantDerivative I E (TangentSpace I)),
  cov.torsion = 0 ↔
    ∀ {X Y : (x : M) → TangentSpace I x} {x : M},
      MDiffAt (T% X) x → MDiffAt (T% 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 232 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceCompleteSpaceCompleteSpaceFiniteDimensionalIsManifold