Theorems · Definition · differential geometry

IsCovariantDerivativeOn.difference

{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] →
                                            {cov cov' : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x} →
                                              {s : Set M} →
                                                IsCovariantDerivativeOn F cov s →
                                                  IsCovariantDerivativeOn F cov' s →
                                                    [CompleteSpace 𝕜] →
                                                      [FiniteDimensional 𝕜 F] →
                                                        [inst_17 : VectorBundle 𝕜 F V] →
                                                          [ContMDiffVectorBundle 1 F V I] →
                                                            (x : M) → V x →L[𝕜] TangentSpace I x →L[𝕜] V x

The difference of two covariant derivatives, as a one-form taking values in the endomorphisms of V.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic

Cited by

1 results in Mathlib

Foundations

Depth 227 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousSMulFiberBundleCompleteSpaceFiniteDimensionalVectorBundleContMDiffVectorBundle