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110f4f3 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import Mathlib
namespace OAI
noncomputable section
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
open Set Filter
open scoped Topology NNReal
open Set Filter
open scoped Topology
open Set Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set
open scoped Topology
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter ContinuousLinearMap InnerProductSpace
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open scoped NNReal
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology
open MeasureTheory
open scoped ContDiff ENNReal
open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory
open scoped Topology ContDiff Manifold Bundle ENNReal
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology Manifold
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Bundle
open scoped Topology Bundle
namespace WeakMTWTransport
variable {B : Type*} [TopologicalSpace B]
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
{E : B → Type*} [TopologicalSpace (TotalSpace F E)]
[∀ x, NormedAddCommGroup (E x)] [∀ x, InnerProductSpace ℝ (E x)]
[FiberBundle F E] [VectorBundle ℝ F E] [IsContinuousRiemannianBundle F E]
lemma continuous_bundle_norm : Continuous (fun z : TotalSpace F E => ‖z.2‖) := by
have h : Continuous (fun z : TotalSpace F E => inner ℝ z.2 z.2) :=
(continuous_id : Continuous (id : TotalSpace F E → _)).inner_bundle continuous_id
simpa only [← norm_eq_sqrt_real_inner] using h.sqrt
lemma isCompact_bundle_disk [CompactSpace B] [T2Space B]
[FiniteDimensional ℝ F] (R : ℝ) :
IsCompact {z : TotalSpace F E | ‖z.2‖ ≤ R} := by
classical
have hsclosed : IsClosed {z : TotalSpace F E | ‖z.2‖ ≤ R} :=
isClosed_le continuous_bundle_norm continuous_const
by_cases hR : R < 0
· have : {z : TotalSpace F E | ‖z.2‖ ≤ R} = ∅ := by
ext z
simp only [mem_ofPred_eq,mem_empty_iff_false,iff_false]
exact not_le.mpr ((hR.trans_le (norm_nonneg _)))
rw [this]
exact isCompact_empty
have hlocal (x : B) : ∃ C : ℝ, 0 < C ∧ ∃ K : Set B,
K ∈ 𝓝 x ∧ IsCompact K ∧ K ⊆ (trivializationAt F E x).baseSet ∧
∀ y ∈ K, ‖(trivializationAt F E x).continuousLinearMapAt ℝ y‖ < C := by
obtain ⟨C,hC,hbound⟩ := eventually_norm_trivializationAt_lt F E x
obtain ⟨K,hK,hKs,hKc⟩ := local_compact_nhds
(inter_mem ((trivializationAt F E x).open_baseSet.mem_nhds
(mem_baseSet_trivializationAt F E x)) hbound)
exact ⟨C,hC,K,hK,hKc,fun y hy => (hKs hy).1,fun y hy => (hKs hy).2⟩
choose C hC K hKn hKc hKb hKC using hlocal
let L (x : B) : Set (TotalSpace F E) :=
(trivializationAt F E x).toOpenPartialHomeomorph.symm ''
(K x ×ˢ Metric.closedBall (0 : F) (C x * R))
have hLc (x : B) : IsCompact (L x) := by
apply ((hKc x).prod (isCompact_closedBall (0 : F) (C x * R))).image_of_continuousOn
apply (trivializationAt F E x).toOpenPartialHomeomorph.continuousOn_symm.mono
intro y hy
change y ∈ (trivializationAt F E x).target
exact (trivializationAt F E x).mem_target.mpr (hKb x hy.1)
obtain ⟨t,ht⟩ := finite_cover_nhds hKn
apply (t.isCompact_biUnion (fun x _ => hLc x)).of_isClosed_subset hsclosed
intro z hz
have hzm : z.1 ∈ ⋃ x ∈ t, K x := by rw [ht]; trivial
obtain ⟨x,hx,hzK⟩ := mem_iUnion₂.mp hzm
apply mem_iUnion₂.mpr ⟨x,hx,?_⟩
have hzb := hKb x hzK
have hzs : z ∈ (trivializationAt F E x).source :=
(trivializationAt F E x).mem_source.mpr hzb
refine ⟨(trivializationAt F E x) z,?_,?_⟩
· refine ⟨?_,?_⟩
· rwa [Trivialization.coe_fst _ hzs]
· rw [Metric.mem_closedBall,dist_zero_right]
have he : (trivializationAt F E x z).2 =
(trivializationAt F E x).continuousLinearMapAt ℝ z.1 z.2 := by
rw [Trivialization.continuousLinearMapAt_apply_of_mem ℝ _ hzb]
rw [he]
calc
_ ≤ ‖(trivializationAt F E x).continuousLinearMapAt ℝ z.1‖ * ‖z.2‖ :=
ContinuousLinearMap.le_opNorm _ _
_ ≤ C x * R := mul_le_mul (hKC x z.1 hzK).le hz (norm_nonneg _) (hC x).le
· exact (trivializationAt F E x).left_inv hzs
end WeakMTWTransport
end
end OAI
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