Theorems · Theorem · differential geometry

Manifold.riemannianEDist_le_pathELength

∀ {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)] [∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)] {x y : M} {a b : ℝ}
  {γ : ℝ → M},
  ContMDiffOn (modelWithCornersSelf ℝ ℝ) I 1 γ (Set.Icc a b) →
    γ a = x → γ b = y → a ≤ b → Manifold.riemannianEDist I x y ≤ Manifold.pathELength I γ a b

The Riemannian edistance is bounded above by the length of any C^1 path from x to y. Here, we express this using a path defined on the whole real line, considered on some interval [a, b].

Defined in

Mathlib.Geometry.Manifold.Riemannian.PathELength

Cited by

4 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceENormENormSMulClass