Theorems · Theorem · global analysis

exists_isMIntegralCurve_iff_exists_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, γ 0 = x ∧ IsMIntegralCurve γ v) ↔ ∀ (a : ℝ), ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Set.Ioo (-a) a)

The existence of a global integral curve is equivalent to the existence of a family of local 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.

Defined in

Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Cited by

1 results in Mathlib

Foundations

Depth 221 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldT2SpaceBoundarylessManifold