| import OAI.Geometry.IsometricImmersion.Metrics.MetricLocality |
| import OAI.Geometry.IsometricImmersion.Metrics.ModelProfile |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Geometry |
| open SmoothLocal.Model |
|
|
| def patchBox : Set Coord := |
| Icc (fun _ => (-(1 / 10 : ℝ))) (fun _ => (1 / 10 : ℝ)) |
|
|
| def centralBox : Set Coord := |
| Icc (fun _ => (-(1 / 5 : ℝ))) (fun _ => (1 / 5 : ℝ)) |
|
|
| theorem patchBox_subset_centralBox : patchBox ⊆ centralBox := by |
| intro p hp |
| constructor <;> intro i <;> linarith [hp.1 i, hp.2 i] |
|
|
| theorem gaussianCurvature_eventuallyEq_of_metric_eventuallyEq |
| {g h : MetricField} {p : Coord} (heq : g =ᶠ[𝓝 p] h) : |
| gaussianCurvature g =ᶠ[𝓝 p] gaussianCurvature h := by |
| filter_upwards [heq.eventuallyEq_nhds] with q hq |
| exact gaussianCurvature_eq_of_eventuallyEq hq |
|
|
| theorem patch_smallCurvature_eventually_model |
| {g0 eta : MetricField} {kappa : ℝ} {U : Set Coord} |
| (hk : 0 < kappa) (hU : IsOpen U) |
| (hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p) |
| (hsupport : tsupport eta ⊆ patchBox) |
| (hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0 + eta) p < -kappa / 2) |
| {p : Coord} (hp : p ∈ U) |
| (hsmall : |gaussianCurvature (g0 + eta) p| ≤ kappa / 8) : |
| gaussianCurvature (g0 + eta) =ᶠ[𝓝 p] modelCurvature kappa := by |
| have hout : p ∉ patchBox := by |
| intro hin |
| have hneg := hcentral p (patchBox_subset_centralBox hin) |
| have hlo := (abs_le.mp hsmall).1 |
| linarith |
| have hts : p ∉ tsupport eta := fun hin => hout (hsupport hin) |
| have hsame := gaussianCurvature_eventuallyEq_of_metric_eventuallyEq |
| (metric_add_eventuallyEq_of_not_mem_tsupport g0 eta hts) |
| have hmodel : gaussianCurvature g0 =ᶠ[𝓝 p] modelCurvature kappa := by |
| filter_upwards [hU.mem_nhds hp] with q hq |
| exact hbackground q hq |
| exact hsame.trans hmodel |
|
|
| theorem patch_smallCurvature_model_firstJet |
| {g0 eta : MetricField} {kappa : ℝ} {U : Set Coord} |
| (hk : 0 < kappa) (hU : IsOpen U) |
| (hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p) |
| (hsupport : tsupport eta ⊆ patchBox) |
| (hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0 + eta) p < -kappa / 2) |
| {p : Coord} (hp : p ∈ U) |
| (hsmall : |gaussianCurvature (g0 + eta) p| ≤ kappa / 8) : |
| gaussianCurvature (g0 + eta) p = modelCurvature kappa p ∧ |
| ∀ i : Fin 2, coordPartial i (gaussianCurvature (g0 + eta)) p = |
| coordPartial i (modelCurvature kappa) p := by |
| have heq := patch_smallCurvature_eventually_model hk hU hbackground hsupport hcentral hp hsmall |
| exact ⟨heq.self_of_nhds, fun i => coordPartial_eq_of_eventuallyEq heq i⟩ |
|
|
| end SmoothLocal.Geometry |
|
|
| end |
|
|
| end OAI |
|
|
