Theorems · Theorem · differential geometry

eventually_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 : Bundle.RiemannianBundle fun x => TangentSpace I x] [inst_6 : IsManifold I 1 M]
  [IsContinuousRiemannianBundle E fun x => TangentSpace I x] (x : M) {c : ENNReal},
  0 < c → ∀ᶠ (y : M) in nhds x, Manifold.riemannianEDist I x y < c

If points are close for the topology, then their Riemannian distance is small.

Defined in

Mathlib.Geometry.Manifold.Riemannian.Basic

Cited by

0 results in Mathlib

Foundations

Depth 282 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceBundle.RiemannianBundleIsManifoldIsContinuousRiemannianBundle