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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 | import OAI.Geometry.IsometricImmersion.Assembly.SupportedNeighborhoods
import OAI.Geometry.IsometricImmersion.Assembly.SupportedMetricJets
import Mathlib.Topology.Closure
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise Distributions
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry
@[simp] theorem perturbationSeminorm_zero (k : ℕ) :
perturbationSeminorm k 0 = 0 := map_zero (perturbationSeminormFamily k)
theorem closure_exists_finite_seminorm_approximation
{T : Type*} [TopologicalSpace T] (f : T → SymmetricPerturbation) (hf : Continuous f)
{A : Set T} {theta : T} (hclosure : theta ∈ closure A)
(n : ℕ) {epsilon : ℝ} (hepsilon : 0 < epsilon) :
∃ eta ∈ A, ∀ k ≤ n, perturbationSeminorm k (f eta - f theta) < epsilon := by
let O : Set T := {eta | ∀ k : Fin (n + 1),
perturbationSeminorm k.val (f eta - f theta) < epsilon}
have hO : IsOpen O := by
change IsOpen {eta | ∀ k : Fin (n + 1),
perturbationSeminorm k.val (f eta - f theta) < epsilon}
simp only [ofPred_forall]
apply isOpen_iInter_of_finite
intro k
exact isOpen_lt ((perturbationSeminorm_continuous k.val).comp
(hf.sub continuous_const)) continuous_const
have htheta : theta ∈ O := by
intro k
simpa only [sub_self, perturbationSeminorm_zero] using hepsilon
obtain ⟨eta, hetaO, hetaA⟩ := mem_closure_iff.mp hclosure O hO htheta
refine ⟨eta, hetaA, ?_⟩
intro k hk
exact hetaO ⟨k, by omega⟩
theorem metricCoefficientCLM_norm_le (i j : Fin 2) :
‖(show (Fin 2 → Fin 2 → ℝ) →L[ℝ] ℝ from metricCoefficientCLM i j)‖ ≤ 1 := by
apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
intro A
rw [one_mul]
exact (norm_le_pi_norm (A i) j).trans (norm_le_pi_norm A i)
theorem perturbation_coefficient_derivative_norm_le
(eta : SymmetricPerturbation) (k : ℕ) (p : Coord) (i j : Fin 2) :
‖iteratedFDeriv ℝ k (fun x => perturbationTensor eta x i j) p‖ ≤
perturbationSeminorm k eta := by
apply ContinuousMultilinearMap.opNorm_le_bound
((norm_nonneg _).trans (perturbation_derivative_norm_le eta k p))
intro v
change ‖iteratedFDeriv ℝ k (fun x => eta.val x i j) p v‖ ≤ _
rw [supportedTensor_coeff_iteratedFDeriv]
calc
_ ≤ ‖iteratedFDeriv ℝ k eta.val p v‖ := Matrix.norm_entry_le_entrywise_sup_norm _
_ ≤ ‖iteratedFDeriv ℝ k eta.val p‖ * ∏ l, ‖v l‖ :=
(iteratedFDeriv ℝ k eta.val p).le_opNorm v
_ ≤ perturbationSeminorm k eta * ∏ l, ‖v l‖ :=
mul_le_mul_of_nonneg_right (perturbation_derivative_norm_le eta k p)
(Finset.prod_nonneg (fun _ _ => norm_nonneg _))
theorem perturbedMetric_sub_eq_tensor_sub (g0 : MetricField) (eta theta : SymmetricPerturbation) :
(fun p => perturbedMetric g0 eta p - perturbedMetric g0 theta p) =
perturbationTensor (eta - theta) := by
funext p
ext i j
change (g0 p i j + perturbationTensor eta p i j) -
(g0 p i j + perturbationTensor theta p i j) =
perturbationTensor eta p i j - perturbationTensor theta p i j
ring
theorem perturbedMetric_coefficient_sub_eq (g0 : MetricField)
(eta theta : SymmetricPerturbation) (i j : Fin 2) :
(fun p => perturbedMetric g0 eta p i j - perturbedMetric g0 theta p i j) =
(fun p => perturbationTensor (eta - theta) p i j) := by
funext p
exact congrFun (congrFun (congrFun (perturbedMetric_sub_eq_tensor_sub g0 eta theta) p) i) j
theorem closure_exists_metric_jet_approximation
{T : Type*} [TopologicalSpace T] (f : T → SymmetricPerturbation) (hf : Continuous f)
(g0 : MetricField) {A : Set T} {theta : T} (hclosure : theta ∈ closure A)
(n : ℕ) {epsilon : ℝ} (hepsilon : 0 < epsilon) :
∃ eta ∈ A,
(∀ k ≤ n, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 (f eta) x -
perturbedMetric g0 (f theta) x) p‖ ≤ epsilon) ∧
(∀ i j : Fin 2, ∀ k ≤ n, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 (f eta) x i j -
perturbedMetric g0 (f theta) x i j) p‖ ≤ epsilon) := by
obtain ⟨eta, heta, hsmall⟩ := closure_exists_finite_seminorm_approximation f hf hclosure n hepsilon
refine ⟨eta, heta, ?_, ?_⟩
· intro k hk p
rw [perturbedMetric_sub_eq_tensor_sub]
exact (perturbation_derivative_norm_le (f eta - f theta) k p).trans (hsmall k hk).le
· intro i j k hk p
rw [perturbedMetric_coefficient_sub_eq]
exact (perturbation_coefficient_derivative_norm_le (f eta - f theta) k p i j).trans (hsmall k hk).le
theorem closure_exists_metric_jet_approximation_with_witness
{T H : Type*} [TopologicalSpace T] (f : T → SymmetricPerturbation) (hf : Continuous f)
(g0 : MetricField) (P : T → H → Prop) {theta : T}
(hclosure : theta ∈ closure {eta | ∃ h : H, P eta h})
(n : ℕ) {epsilon : ℝ} (hepsilon : 0 < epsilon) :
∃ (eta : T) (h : H), P eta h ∧
(∀ k ≤ n, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 (f eta) x -
perturbedMetric g0 (f theta) x) p‖ ≤ epsilon) ∧
(∀ i j : Fin 2, ∀ k ≤ n, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 (f eta) x i j -
perturbedMetric g0 (f theta) x i j) p‖ ≤ epsilon) := by
obtain ⟨eta, ⟨h, hP⟩, hT, hC⟩ :=
closure_exists_metric_jet_approximation f hf g0 hclosure n hepsilon
exact ⟨eta, h, hP, hT, hC⟩
end SmoothLocal.Perturbation
end
end OAI
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