Theorems · Theorem · differential geometry

CovariantDerivative.ContMDiffCovariantDerivative.affineCombination

∀ {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] [inst_15 : IsManifold I 1 M]
  [inst_16 : VectorBundle 𝕜 F V] (cov cov' : CovariantDerivative I F V) {f : M → 𝕜} {n : WithTop ℕ∞},
  ContMDiff I (modelWithCornersSelf 𝕜 𝕜) n f →
    cov.ContMDiffCovariantDerivative n →
      cov'.ContMDiffCovariantDerivative n → (cov.affineCombination cov' f).ContMDiffCovariantDerivative n

An affine combination of two C^k connections is a C^k connection.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic

Cited by

1 results in Mathlib

Foundations

Depth 219 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousSMulFiberBundleIsManifoldVectorBundle