Theorems · Theorem · differential geometry

Manifold.pathELength_comp_of_antitoneOn

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {H : Type u_2} [inst_2 : TopologicalSpace H]
  {I : ModelWithCorners ℝ E H} {M : Type u_3} [inst_3 : TopologicalSpace M] [inst_4 : ChartedSpace H M]
  [inst_5 : (x : M) → ENorm (TangentSpace I x)] {a b : ℝ} {γ : ℝ → M} [∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)]
  {f : ℝ → ℝ},
  a ≤ b →
    AntitoneOn f (Set.Icc a b) →
      DifferentiableOn ℝ f (Set.Icc a b) →
        MDiff[Set.Icc (f b) (f a)] γ → Manifold.pathELength I (γ ∘ f) a b = Manifold.pathELength I γ (f b) (f a)

The length of a path in a manifold is invariant under an antitone reparametrization.

Defined in

Mathlib.Geometry.Manifold.Riemannian.PathELength

Cited by

1 results in Mathlib

Foundations

Depth 280 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceENormENormSMulClass