Theorems · Theorem · global analysis

eqOn_abs_add_one_of_isMIntegralCurveOn_Ioo

∀ {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 : IsManifold I 1 M] [T2Space M] {v : (x : M) → TangentSpace I x} [BoundarylessManifold I M],
  (ContMDiff I I.tangent 1 fun x => ⟨x, v x⟩) →
    ∀ {x : M} (γ : ℝ → ℝ → M),
      (∀ (a : ℝ), γ a 0 = x) →
        (∀ a > 0, IsMIntegralCurveOn (γ a) v (Set.Ioo (-a) a)) →
          ∀ {a : ℝ}, Set.EqOn (fun t => γ (|t| + 1) t) (γ a) (Set.Ioo (-a) a)

For a family of integral curves γ : ℝ → ℝ → M with the same starting point γ 0 = x such that each γ a is defined on Ioo (-a) a, the global curve γ_ext := fun t ↦ γ (|t| + 1) t agrees with each γ a on Ioo (-a) a. This will help us show that γ_ext is a global integral curve.

Defined in

Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Cited by

1 results in Mathlib

Foundations

Depth 219 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldT2SpaceBoundarylessManifold