Theorems · Theorem · global analysis

isMIntegralCurveOn_Ioo_eqOn_of_contMDiff

∀ {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] {γ γ' : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ} [T2Space M] {a b : ℝ},
  t₀ ∈ Set.Ioo a b →
    (∀ t ∈ Set.Ioo a b, I.IsInteriorPoint (γ t)) →
      (ContMDiff I I.tangent 1 fun x => ⟨x, v x⟩) →
        IsMIntegralCurveOn γ v (Set.Ioo a b) →
          IsMIntegralCurveOn γ' v (Set.Ioo a b) → γ t₀ = γ' t₀ → Set.EqOn γ γ' (Set.Ioo a b)

Integral curves are unique on open intervals. If a $C^1$ vector field v admits two integral curves γ γ' : ℝ → M on some open interval Ioo a b, and γ t₀ = γ' t₀ for some t ∈ Ioo a b, then γ and γ' agree on Ioo a b.

Defined in

Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique

Cited by

2 results in Mathlib

Foundations

Depth 216 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldT2Space