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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 | import OAI.Geometry.IsometricImmersion.Assembly.TensorSeriesLocality
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Geometry
theorem tensorPatchSum_eq_single_on_disjoint_carrier
(η : ℕ → MetricField) (C : ℕ → Set Coord)
(hsupp : ∀ j, tsupport (η j) ⊆ C j)
(hdisjoint : Pairwise (fun i j => Disjoint (C i) (C j)))
(i : ℕ) {p : Coord} (hp : p ∈ C i) : tensorPatchSum η p = η i p := by
change (∑' j, η j p) = η i p
apply tsum_eq_single i
intro j hji
by_contra hne
exact Set.disjoint_left.mp (hdisjoint hji)
(hsupp j (subset_tsupport (η j) hne)) hp
theorem tensorPatchSum_zero_off_carriers
(η : ℕ → MetricField) (C : ℕ → Set Coord)
(hsupp : ∀ j, tsupport (η j) ⊆ C j)
{p : Coord} (hp : ∀ j, p ∉ C j) : tensorPatchSum η p = 0 := by
have hz (j : ℕ) : η j p = 0 := by
by_contra hne
exact hp j (hsupp j (subset_tsupport (η j) hne))
change (∑' j, η j p) = 0
simp only [hz, tsum_zero]
theorem tensorPatchSum_posDef_of_disjoint
(η : ℕ → MetricField) (C : ℕ → Set Coord)
(hsupp : ∀ j, tsupport (η j) ⊆ C j)
(hdisjoint : Pairwise (fun i j => Disjoint (C i) (C j)))
{g : MetricField} {U : Set Coord}
(hg : ∀ p ∈ U, (g p).PosDef)
(hlocal : ∀ j, ∀ p ∈ U ∩ C j, ((g + η j) p).PosDef) :
∀ p ∈ U, ((g + tensorPatchSum η) p).PosDef := by
intro p hp
by_cases hex : ∃ j, p ∈ C j
· obtain ⟨j, hj⟩ := hex
change (g p + tensorPatchSum η p).PosDef
rw [tensorPatchSum_eq_single_on_disjoint_carrier η C hsupp hdisjoint j hj]
exact hlocal j p ⟨hp, hj⟩
· have hout : ∀ j, p ∉ C j := fun j hj => hex ⟨j, hj⟩
change (g p + tensorPatchSum η p).PosDef
rw [tensorPatchSum_zero_off_carriers η C hsupp hout, add_zero]
exact hg p hp
theorem tensorPatchSum_smoothPositive_of_disjoint
(η : ℕ → MetricField) (C : ℕ → Set Coord) (ε : ℝ)
(hη : ∀ j, ContDiff ℝ ∞ (η j)) (hC : ∀ j, IsCompact (C j))
(hsupp : ∀ j, tsupport (η j) ⊆ C j)
(hdisjoint : Pairwise (fun i j => Disjoint (C i) (C j)))
(hbound : ∀ j k : ℕ, k ≤ j → ∀ p : Coord,
‖iteratedFDeriv ℝ k (η j) p‖ ≤ patchSeriesWeight ε j)
{g : MetricField} {U : Set Coord} (hg : SmoothPositiveOn g U)
(hlocal : ∀ j, ∀ p ∈ U ∩ C j, ((g + η j) p).PosDef) :
SmoothPositiveOn (g + tensorPatchSum η) U := by
have hc (j : ℕ) : HasCompactSupport (η j) :=
(hC j).of_isClosed_subset (isClosed_tsupport (η j)) (hsupp j)
exact smoothPositive_metric_add_of_tensor_contDiff hg
(tensorPatchSum_contDiff η ε hη hc hbound)
(tensorPatchSum_posDef_of_disjoint η C hsupp hdisjoint hg.2 hlocal)
theorem tensorPatchSum_preserves_disk_of_disjoint
(η : ℕ → MetricField) (C : ℕ → Set Coord) (ε : ℝ)
(hη : ∀ j, ContDiff ℝ ∞ (η j)) (hC : ∀ j, IsCompact (C j))
(hsupp : ∀ j, tsupport (η j) ⊆ C j)
(hdisjoint : Pairwise (fun i j => Disjoint (C i) (C j)))
(hbound : ∀ j k : ℕ, k ≤ j → ∀ p : Coord,
‖iteratedFDeriv ℝ k (η j) p‖ ≤ patchSeriesWeight ε j)
{g : MetricField} {U : Set Coord} (hg : SmoothPositiveOn g U) (hU : IsOpen U)
(hlocal : ∀ j, ∀ p ∈ U ∩ C j, ((g + η j) p).PosDef)
(n : ℕ) (hDU : accumulatingDisk n ⊆ U)
(havoid : ∀ j, Disjoint (C j) (accumulatingDisk n)) :
∀ p ∈ accumulatingDisk n,
(∀ k : ℕ, iteratedFDeriv ℝ k (tensorPatchSum η) p = 0) ∧
(∀ ds : List (Fin 2), ∀ a b : Fin 2,
iteratedCoordPartial ds (fun q => (g + tensorPatchSum η) q a b) p =
iteratedCoordPartial ds (fun q => g q a b) p) ∧
gaussianCurvature (g + tensorPatchSum η) p = gaussianCurvature g p := by
have hc (j : ℕ) : HasCompactSupport (η j) :=
(hC j).of_isClosed_subset (isClosed_tsupport (η j)) (hsupp j)
exact tensorPatchSum_preserves_closed_disk η ε hη hc hbound hg hU
(tensorPatchSum_posDef_of_disjoint η C hsupp hdisjoint hg.2 hlocal) n hDU
(fun j => (havoid j).mono_left (hsupp j))
end SmoothLocal.Geometry
end
end OAI
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