Theorems · Theorem · differential geometry

IsCovariantDerivativeOn.affine_combination

∀ {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] {F : Type u_5} [inst_6 : NormedAddCommGroup F]
  [inst_7 : NormedSpace 𝕜 F] {V : M → Type u_6} [inst_8 : TopologicalSpace (Bundle.TotalSpace F V)]
  [inst_9 : (x : M) → AddCommGroup (V x)] [inst_10 : (x : M) → Module 𝕜 (V x)]
  [inst_11 : (x : M) → TopologicalSpace (V x)] [inst_12 : ∀ (x : M), IsTopologicalAddGroup (V x)]
  [inst_13 : ∀ (x : M), ContinuousSMul 𝕜 (V x)] [inst_14 : FiberBundle F V] {s : Set M}
  {cov : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x},
  IsCovariantDerivativeOn F cov s →
    ∀ {cov' : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x},
      IsCovariantDerivativeOn F cov' s →
        ∀ (g : M → 𝕜), IsCovariantDerivativeOn F (fun σ => g • cov σ + (1 - g) • cov' σ) s

An affine combination of covariant derivatives is a covariant derivative.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic

Cited by

0 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousSMulFiberBundle