Theorems · Theorem · differential geometry

Manifold.exists_lt_locally_constant_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 : M), ENormSMulClass ℝ (TangentSpace I x)] {x y : M} {r : ENNReal}
  {a b : ℝ},
  Manifold.riemannianEDist I x y < r →
    a < b →
      ∃ γ,
        γ a = x ∧
          γ b = y ∧
            ContMDiff (modelWithCornersSelf ℝ ℝ) I 1 γ ∧
              Manifold.pathELength I γ a b < r ∧ (γ =ᶠ[nhds a] fun x_1 => x) ∧ γ =ᶠ[nhds b] fun x => y

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 an arbitrary interval [a, b] with a < b, and moreover we ensure that the path is locally constant around a and b, which is convenient for gluing purposes.

Defined in

Mathlib.Geometry.Manifold.Riemannian.PathELength

Cited by

3 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceENormENormSMulClass