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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 | import OAI.Geometry.IsometricImmersion.Assembly.EnumeratedPatchTensors
import OAI.Geometry.IsometricImmersion.Obstructions.PatchAdmissibility
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Geometry
open SmoothLocal.Perturbation
def assembledPatchTensor (eta : PatchAddress → SymmetricPerturbation) : MetricField :=
tensorPatchSum (enumeratedPatchTensor eta)
def assembledPatchMetric (g : MetricField) (eta : PatchAddress → SymmetricPerturbation) : MetricField :=
g + assembledPatchTensor eta
def localPatchMetric (g : MetricField) (eta : PatchAddress → SymmetricPerturbation)
(a : PatchAddress) : MetricField :=
affinePullbackMetric g (patchAddressCenter a) (patchAddressMatrix a) + perturbationTensor (eta a)
theorem assembledPatchTensor_eq_single (eta : PatchAddress → SymmetricPerturbation)
(a : PatchAddress) {p : Coord} (hp : p ∈ patchAddressCarrier a) :
assembledPatchTensor eta p = patchAddressTensor eta a p := by
have hh := tensorPatchSum_eq_single_on_disjoint_carrier
(enumeratedPatchTensor eta) enumeratedPatchCarrier
(enumeratedPatchTensor_tsupport_subset eta) enumeratedPatchCarriers_pairwise_disjoint
(patchTensorIndex a) (by simpa only [enumeratedPatchCarrier_index] using hp)
simpa only [assembledPatchTensor,enumeratedPatchTensor_index] using hh
theorem assembledPatchTensor_zero_off_carriers (eta : PatchAddress → SymmetricPerturbation)
{p : Coord} (hp : ∀ a : PatchAddress, p ∉ patchAddressCarrier a) : assembledPatchTensor eta p = 0 := by
apply tensorPatchSum_zero_off_carriers (enumeratedPatchTensor eta) enumeratedPatchCarrier
(enumeratedPatchTensor_tsupport_subset eta)
intro j
cases h : patchTensorAddress j with
| none => simp [enumeratedPatchCarrier,h]
| some a => simpa only [enumeratedPatchCarrier,h] using hp a
theorem assembledPatchMetric_pullback_eq_on_closedPatchBox
(g : MetricField) (eta : PatchAddress → SymmetricPerturbation) (a : PatchAddress)
{q : Coord} (hq : q ∈ closedPatchBox) :
affinePullbackMetric (assembledPatchMetric g eta) (patchAddressCenter a) (patchAddressMatrix a) q =
localPatchMetric g eta a q := by
have hp : affineCoordinates (patchAddressCenter a) (patchAddressMatrix a) q ∈ patchAddressCarrier a :=
⟨q,hq,rfl⟩
calc
_ = affinePullbackMetric (g + patchAddressTensor eta a)
(patchAddressCenter a) (patchAddressMatrix a) q := by
unfold affinePullbackMetric assembledPatchMetric
simp only [Pi.add_apply,assembledPatchTensor_eq_single eta a hp]
_ = _ := congrFun (affinePullback_add_pushforward g (perturbationTensor (eta a))
(patchAddressCenter a) (patchAddressMatrix a) (patchAddressMatrix_isUnit a)) q
theorem assembledPatchMetric_pullback_eventuallyEq
(g : MetricField) (eta : PatchAddress → SymmetricPerturbation) (a : PatchAddress)
{q : Coord} (hq : ‖q‖ < 4) :
affinePullbackMetric (assembledPatchMetric g eta) (patchAddressCenter a) (patchAddressMatrix a)
=ᶠ[𝓝 q] localPatchMetric g eta a := by
have hball : q ∈ Metric.ball (0 : Coord) 4 := by
simpa only [Metric.mem_ball,dist_zero_right] using hq
filter_upwards [Metric.isOpen_ball.mem_nhds hball] with p hp
apply assembledPatchMetric_pullback_eq_on_closedPatchBox g eta a
apply (mem_closedPatchBox_iff_norm_le p).mpr
exact (by simpa only [Metric.mem_ball,dist_zero_right] using hp : ‖p‖ < 4).le
theorem assembledPatchTensor_contDiff {epsilon : ℝ} (he : 0 < epsilon)
(eta : PatchAddress → SymmetricPerturbation)
(hsmall : ∀ a, eta a ∈ patchTensorBudgetNeighborhood epsilon a) :
ContDiff ℝ ∞ (assembledPatchTensor eta) :=
tensorPatchSum_contDiff _ epsilon (enumeratedPatchTensor_contDiff eta)
(enumeratedPatchTensor_hasCompactSupport eta) (enumeratedPatchTensor_diagonal_bound he eta hsmall)
theorem assembledPatchTensor_summable {epsilon : ℝ} (he : 0 < epsilon)
(eta : PatchAddress → SymmetricPerturbation)
(hsmall : ∀ a, eta a ∈ patchTensorBudgetNeighborhood epsilon a) (p : Coord) :
Summable (fun j => enumeratedPatchTensor eta j p) :=
tensorPatchSum_summable _ epsilon (enumeratedPatchTensor_diagonal_bound he eta hsmall) p
theorem assembledPatchTensor_all_jets_zero_at_origin {epsilon : ℝ} (he : 0 < epsilon)
(eta : PatchAddress → SymmetricPerturbation)
(hsmall : ∀ a, eta a ∈ patchTensorBudgetNeighborhood epsilon a) (k : ℕ) :
iteratedFDeriv ℝ k (assembledPatchTensor eta) (0 : Coord) = 0 := by
apply tensorPatchSum_jets_zero_off_each_support _ epsilon (enumeratedPatchTensor_contDiff eta)
(enumeratedPatchTensor_hasCompactSupport eta) (enumeratedPatchTensor_diagonal_bound he eta hsmall)
intro j hj
exact zero_not_mem_enumeratedPatchCarrier j (enumeratedPatchTensor_tsupport_subset eta j hj)
theorem assembledPatchTensor_all_jets_zero_on_disk {epsilon : ℝ} (he : 0 < epsilon)
(eta : PatchAddress → SymmetricPerturbation)
(hsmall : ∀ a, eta a ∈ patchTensorBudgetNeighborhood epsilon a)
(n : ℕ) {p : Coord} (hp : p ∈ accumulatingDisk n) (k : ℕ) :
iteratedFDeriv ℝ k (assembledPatchTensor eta) p = 0 := by
apply tensorPatchSum_jets_zero_off_each_support _ epsilon (enumeratedPatchTensor_contDiff eta)
(enumeratedPatchTensor_hasCompactSupport eta) (enumeratedPatchTensor_diagonal_bound he eta hsmall)
intro j hj
exact Set.disjoint_left.mp (enumeratedPatchCarrier_disjoint_disk j n)
(enumeratedPatchTensor_tsupport_subset eta j hj) hp
theorem assembledPatchTensor_jets_eq_single_on_carrier {epsilon : ℝ} (he : 0 < epsilon)
(eta : PatchAddress → SymmetricPerturbation)
(hsmall : ∀ a, eta a ∈ patchTensorBudgetNeighborhood epsilon a)
(a : PatchAddress) {p : Coord} (hp : p ∈ patchAddressCarrier a) (k : ℕ) :
iteratedFDeriv ℝ k (assembledPatchTensor eta) p = iteratedFDeriv ℝ k (patchAddressTensor eta a) p := by
have hi : p ∈ enumeratedPatchCarrier (patchTensorIndex a) := by
simpa only [enumeratedPatchCarrier_index] using hp
have hjets := tensorPatchSum_jets_eq_single _ epsilon (enumeratedPatchTensor_contDiff eta)
(enumeratedPatchTensor_hasCompactSupport eta) (enumeratedPatchTensor_diagonal_bound he eta hsmall)
(patchTensorIndex a) (p := p)
(fun j hj hmem => Set.disjoint_left.mp (enumeratedPatchCarriers_pairwise_disjoint hj)
(enumeratedPatchTensor_tsupport_subset eta j hmem) hi) k
simpa only [assembledPatchTensor,enumeratedPatchTensor_index] using hjets
theorem assembledPatchMetric_posDef
{g : MetricField} {U : Set Coord} (hg : ∀ p ∈ U, (g p).PosDef)
(eta : PatchAddress → SymmetricPerturbation)
(hlocal : ∀ a, ∀ p ∈ U ∩ patchAddressCarrier a, ((g + patchAddressTensor eta a) p).PosDef) :
∀ p ∈ U, (assembledPatchMetric g eta p).PosDef := by
intro p hp
by_cases hex : ∃ a, p ∈ patchAddressCarrier a
· obtain ⟨a,ha⟩ := hex
change (g p + assembledPatchTensor eta p).PosDef
rw [assembledPatchTensor_eq_single eta a ha]
exact hlocal a p ⟨hp,ha⟩
· have hout : ∀ a, p ∉ patchAddressCarrier a := fun a ha => hex ⟨a,ha⟩
change (g p + assembledPatchTensor eta p).PosDef
rw [assembledPatchTensor_zero_off_carriers eta hout,add_zero]
exact hg p hp
theorem assembledPatchMetric_smoothPositive {epsilon : ℝ} (he : 0 < epsilon)
{g : MetricField} {U : Set Coord} (hg : SmoothPositiveOn g U)
(eta : PatchAddress → SymmetricPerturbation)
(hsmall : ∀ a, eta a ∈ patchTensorBudgetNeighborhood epsilon a)
(hlocal : ∀ a, ∀ p ∈ U ∩ patchAddressCarrier a, ((g + patchAddressTensor eta a) p).PosDef) :
SmoothPositiveOn (assembledPatchMetric g eta) U :=
smoothPositive_metric_add_of_tensor_contDiff hg (assembledPatchTensor_contDiff he eta hsmall)
(assembledPatchMetric_posDef hg.2 eta hlocal)
theorem assembledPatchMetric_curvature_on_disk {epsilon : ℝ} (he : 0 < epsilon)
{g : MetricField} {U : Set Coord} (hg : SmoothPositiveOn g U) (hU : IsOpen U)
(eta : PatchAddress → SymmetricPerturbation)
(hsmall : ∀ a, eta a ∈ patchTensorBudgetNeighborhood epsilon a)
(hlocal : ∀ a, ∀ p ∈ U ∩ patchAddressCarrier a, ((g + patchAddressTensor eta a) p).PosDef)
(n : ℕ) {p : Coord} (hp : p ∈ accumulatingDisk n) (hpU : p ∈ U) :
gaussianCurvature (assembledPatchMetric g eta) p = gaussianCurvature g p :=
gaussianCurvature_add_eq_of_zero_tensor_jets hg hU hpU
(assembledPatchTensor_contDiff he eta hsmall) (assembledPatchMetric_posDef hg.2 eta hlocal)
(assembledPatchTensor_all_jets_zero_on_disk he eta hsmall n hp)
end SmoothLocal.Geometry
end
end OAI
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