Theorems · Theorem · global analysis

exists_isMIntegralCurve_of_isMIntegralCurveOn

∀ {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] [BoundarylessManifold I M] {v : (x : M) → TangentSpace I x},
  (ContMDiff I I.tangent 1 fun x => ⟨x, v x⟩) →
    ∀ {ε : ℝ},
      0 < ε →
        (∀ (x : M), ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Set.Ioo (-ε) ε)) →
          ∀ (x : M), ∃ γ, γ 0 = x ∧ IsMIntegralCurve γ v

If there exists ε > 0 such that the local integral curve at each point x : M is defined at least on an open interval Ioo (-ε) ε, then every point on M has a global integral curve passing through it. See Lemma 9.15, [J.M. Lee (2012)][lee2012].

Defined in

Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Cited by

0 results in Mathlib

Foundations

Depth 222 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldT2SpaceBoundarylessManifold