Theorems · Inductive type · differential geometry

IsCovariantDerivativeOn

{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] →
                          [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)] →
                                      [∀ (x : M), IsTopologicalAddGroup (V x)] →
                                        [∀ (x : M), ContinuousSMul 𝕜 (V x)] →
                                          [FiberBundle F V] →
                                            (((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x) →
                                              optParam (Set M) Set.univ → Prop

A function from sections of a vector bundle V on a manifold M to sections of $Hom(TM, E)$ is a covariant derivative over a set s in M if it is additive and satisfies the Leibniz rule when applied to sections that are differentiable at a point of s. Caution, the argument order is nonstandard: cov σ x (X x) corresponds to ∇_X σ x on paper.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic

Cited by

24 results in Mathlib

Foundations

Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousSMulFiberBundle