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2fca478 | 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 | import Mathlib
import OAI.Geometry.PrescribedPotential.AnalyticSupport
namespace OAI
/-! Atlas Localization. -/
section
noncomputable section
open Set Filter Topology Manifold IsManifold
open scoped ContDiff
namespace Anticanonical.ComplexAtlas
variable {d : ℕ} {X : Type*} [TopologicalSpace X] (A : ComplexAtlas d X)
@[instance_reducible] def chartedSpace : ChartedSpace (Coordinates d) X where
atlas := Set.range A.chart
chartAt x := A.chart (A.covers x).choose
mem_chart_source x := (A.covers x).choose_spec
chart_mem_atlas point := ⟨(A.covers point).choose, rfl⟩
lemma real_smooth_transition (i j : Fin A.count) :
ContDiffOn ℝ ∞ (A.transition i j) (coordinateOverlap (A.chart i) (A.chart j)) := by
intro z hz
exact ((ProjectiveKaehler.analyticOnNhd_pi_of_differentiableOn
(A.isOpen_coordinateOverlap i j) (A.holomorphic i j) z hz).contDiffAt.restrict_scalars ℝ).contDiffWithinAt
theorem isManifold : letI := A.chartedSpace; IsManifold 𝓘(ℝ, Coordinates d) ∞ X := by
let := A.chartedSpace
apply isManifold_of_contDiffOn
rintro e e' ⟨i, rfl⟩ ⟨j, rfl⟩
simpa only [modelWithCornersSelf_coe, modelWithCornersSelf_coe_symm,
Function.id_comp, Function.comp_id, Set.preimage_id, Set.range_id, Set.inter_univ, OpenPartialHomeomorph.coe_trans,
OpenPartialHomeomorph.trans_source, OpenPartialHomeomorph.symm_source,
transition, coordinateOverlap] using
A.real_smooth_transition i j
theorem contMDiff_iff (f : X → ℝ) : letI := A.chartedSpace;
ContMDiff 𝓘(ℝ, Coordinates d) 𝓘(ℝ) ∞ f ↔
∀ i, ContDiffOn ℝ ∞ (f ∘ (A.chart i).symm) (A.chart i).target := by
let := A.chartedSpace
let := A.isManifold
constructor
· intro hf i
have he : A.chart i ∈ maximalAtlas 𝓘(ℝ, Coordinates d) ∞ X :=
(contDiffGroupoid ∞ 𝓘(ℝ, Coordinates d)).subset_maximalAtlas ⟨i, rfl⟩
exact (hf.comp_contMDiffOn (contMDiffOn_symm_of_mem_maximalAtlas he)).contDiffOn
· intro hf x
rw [contMDiffAt_iff_source]
have hchart : chartAt (Coordinates d) x = A.chart (A.covers x).choose := rfl
have hx := (A.chart (A.covers x).choose).mapsTo (A.covers x).choose_spec
have hh := ((hf (A.covers x).choose).contDiffAt
((A.chart (A.covers x).choose).open_target.mem_nhds hx)).contMDiffAt
simpa only [extChartAt_coe_symm, extChartAt_coe, hchart, modelWithCornersSelf_coe,
modelWithCornersSelf_coe_symm, Function.comp_id, Function.id_comp,
Set.range_id, contMDiffWithinAt_univ, chartAt, chartedSpace] using hh
theorem exists_smooth_partition [T2Space X] [CompactSpace X] :
∃ ρ : Fin A.count → SourceSmooth.SmoothRealFunction A,
(∀ i x, 0 ≤ (ρ i).value x) ∧
(∀ x, ∑ i, (ρ i).value x = 1) ∧
(∀ i, tsupport (ρ i).value ⊆ (A.chart i).source) := by
let := A.chartedSpace
let := A.isManifold
obtain ⟨ρ, hρ⟩ := SmoothPartitionOfUnity.exists_isSubordinate
𝓘(ℝ, Coordinates d) (s := (Set.univ : Set X)) isClosed_univ
(fun i => (A.chart i).source) (fun i => (A.chart i).open_source)
(by intro x _; obtain ⟨i, hi⟩ := A.covers x; exact Set.mem_iUnion.mpr ⟨i, hi⟩)
refine ⟨fun i => ⟨ρ i, (A.contMDiff_iff (ρ i)).mp (ρ i).contMDiff⟩, ?_, ?_, ?_⟩
· exact fun i x => ρ.nonneg i x
· intro x
simpa only [finsum_eq_sum_of_fintype] using ρ.sum_eq_one (Set.mem_univ x)
· exact hρ
end Anticanonical.ComplexAtlas
end
end
end OAI
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