Theorems · Theorem · global analysis

isMIntegralCurveAt_eventuallyEq_of_contMDiffAt

∀ {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},
  I.IsInteriorPoint (γ t₀) →
    ContMDiffAt I I.tangent 1 (fun x => ⟨x, v x⟩) (γ t₀) →
      IsMIntegralCurveAt γ v t₀ → IsMIntegralCurveAt γ' v t₀ → γ t₀ = γ' t₀ → γ =ᶠ[nhds t₀] γ'

Local integral curves are unique. If a $C^1$ vector field v admits two local integral curves γ γ' : ℝ → M at t₀ with γ t₀ = γ' t₀, then γ and γ' agree on some open interval containing t₀.

Defined in

Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique

Cited by

2 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifold