Theorems · Theorem · differential geometry

Manifold.riemannianEDist_def

∀ {E : Type u_4} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {H : Type u_5} [inst_2 : TopologicalSpace H]
  (I : ModelWithCorners ℝ E H) {M : Type u_6} [inst_3 : TopologicalSpace M] [inst_4 : ChartedSpace H M]
  [inst_5 : (x : M) → ENorm (TangentSpace I x)] (x y : M),
  Manifold.riemannianEDist I x y =
    ⨅ γ,
      ⨅ (_ : ContMDiff (modelWithCornersEuclideanHalfSpace 1) I 1 ⇑γ), ∫⁻ (x_2 : ↑unitInterval), ‖(mfderiv% ⇑γ x_2) 1‖ₑ

Defined in

Mathlib.Geometry.Manifold.Riemannian.PathELength

Cited by

2 results in Mathlib

Foundations

Depth 248 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceENorm