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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | import OAI.Geometry.IsometricImmersion.Flows.FlowInverse
import Mathlib.Topology.IsLocalHomeomorph
namespace OAI
noncomputable section
open Set Filter MeasureTheory Function
open scoped ContDiff Topology Interval
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ}
theorem exists_global_triangularFlow_chart
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
(hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
Y s t ∈ Icc (-3 : ℝ) 3)
(hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t) :
∃ e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ),
e.source = pairRectangle 2 (-2) 2 ∧
e.target = triangularFlow Y '' pairRectangle 2 (-2) 2 ∧
(e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y ∧
ContDiffOn ℝ ∞ e e.source ∧ ContDiffOn ℝ ∞ e.symm e.target := by
let D := pairRectangle 2 (-2) 2
have hD : IsOpen D := pairRectangle_isOpen 2 (-2) 2
have hTF : ContDiffOn ℝ ∞ (triangularFlow Y) D :=
contDiffOn_fst.prodMk (cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode)
have hinj : InjOn (triangularFlow Y) D := triangularFlow_injOn hq hU hSU hY hrange hstart hode
have hlocal : IsLocalHomeomorphOn (triangularFlow Y) D := by
intro x hx
obtain ⟨e, he, hsub, heq, hf, hi⟩ :=
exists_triangularFlow_smooth_chart hq hU hSU hY hrange hstart hode hx
exact ⟨e, he, heq.symm⟩
have hopen : IsOpenMap (D.domRestrict (triangularFlow Y)) := by
apply IsOpenMap.of_nhds_le
intro x
have he : (𝓝 x).map (D.domRestrict (triangularFlow Y)) =
𝓝 (triangularFlow Y x) := by
change (𝓝 x).map ((triangularFlow Y) ∘ (Subtype.val : D → ℝ × ℝ)) = _
rw [← Filter.map_map, hD.isOpenEmbedding_subtypeVal.map_nhds_eq,
hlocal.map_nhds_eq x.2]
exact he.ge
let pe := hinj.toPartialEquiv (triangularFlow Y) D
let e := OpenPartialHomeomorph.ofContinuousOpenRestrict pe hTF.continuousOn hopen hD
have heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y := rfl
have hsource : e.source = D := rfl
refine ⟨e, hsource, rfl, heq, ?_, ?_⟩
· rw [heq, hsource]
exact hTF
· intro y hy
have hx : e.symm y ∈ D := by
rw [← hsource]
exact e.map_target hy
obtain ⟨A, hA⟩ := triangularFlow_hasFDerivAt_equiv hq hU hSU hY hrange hstart hode hx
have hAe : HasFDerivAt e (A : (ℝ × ℝ) →L[ℝ] (ℝ × ℝ)) (e.symm y) := by
rw [heq]
exact hA
have hfe : ContDiffAt ℝ ∞ e (e.symm y) := by
rw [heq]
exact hTF.contDiffAt (hD.mem_nhds hx)
exact (e.contDiffAt_symm hy hAe hfe).contDiffWithinAt
def flowCap (Y : ℝ → ℝ → ℝ) (b : ℝ) : Set (ℝ × ℝ) :=
triangularFlow Y '' pairRectangle 2 (-2) b
theorem flowCap_mono (Y : ℝ → ℝ → ℝ) {b1 b2 : ℝ} (hb : b1 ≤ b2) :
flowCap Y b1 ⊆ flowCap Y b2 := by
apply image_mono
intro p hp
exact ⟨hp.1, hp.2.1, hp.2.2.trans_le hb⟩
theorem flowCap_isOpen_of_chart {e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)}
(hsource : e.source = pairRectangle 2 (-2) 2)
(heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y)
{b : ℝ} (hb : b ≤ 2) : IsOpen (flowCap Y b) := by
let C := pairRectangle 2 (-2) b
have hC : IsOpen C := pairRectangle_isOpen 2 (-2) b
have hsub : C ⊆ e.source := by
rw [hsource]
intro p hp
exact ⟨hp.1, hp.2.1, hp.2.2.trans_le hb⟩
let eb := e.restrOpen C hC
have hes : eb.source = C := by
change e.source ∩ C = C
exact inter_eq_right.mpr hsub
have het : eb.target = flowCap Y b := by
rw [← eb.image_source_eq_target, hes]
change e '' C = triangularFlow Y '' C
rw [heq]
rw [← het]
exact eb.open_target
end SmoothLocal.Flow
end
end OAI
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