Theorems · Theorem · differential geometry

Manifold.exists_lt_of_riemannianEDist_lt

∀ {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 y : M} {r : ENNReal},
  Manifold.riemannianEDist I x y < r →
    ∃ γ,
      γ 0 = x ∧ γ 1 = y ∧ ContMDiffOn (modelWithCornersSelf ℝ ℝ) I 1 γ (Set.Icc 0 1) ∧ Manifold.pathELength I γ 0 1 < r

If some r is strictly larger than the Riemannian edistance between two points, there exists a path between these two points of length < r. Here, we get such a path on [0, 1]. For a more precise version giving locally constant paths around the endpoints, see exists_lt_locally_constant_of_riemannianEDist_lt

Defined in

Mathlib.Geometry.Manifold.Riemannian.PathELength

Cited by

1 results in Mathlib

Foundations

Depth 251 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceENorm