| import OAI.Geometry.IsometricImmersion.Assembly.InitialAssemblyMetric |
| import OAI.Geometry.IsometricImmersion.Assembly.EnumeratedPatchTensors |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
| open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise |
|
|
| namespace SmoothLocal.Geometry |
| open SmoothLocal.Perturbation SmoothLocal.Model |
|
|
| def patchAddressInitialMetric (a : PatchAddress) : MetricField := |
| affinePullbackMetric initialAssemblyMetric (patchAddressCenter a) (patchAddressMatrix a) |
|
|
| def patchAddressAmplitude (a : PatchAddress) : ℝ := |
| initialPatchAmplitude a.1 a.2.1 a.2.2.1 a.2.2.2.val a.2.2.2.property |
|
|
| def patchAddressModelDomain : Set Coord := Metric.ball 0 (13/4) |
|
|
| theorem patchAddressAmplitude_pos (a : PatchAddress) : 0 < patchAddressAmplitude a := |
| initialPatchAmplitude_pos a.1 a.2.1 a.2.2.1 a.2.2.2.val a.2.2.2.property |
|
|
| theorem patchAddressModelDomain_isOpen : IsOpen patchAddressModelDomain := Metric.isOpen_ball |
|
|
| theorem modelSquare_subset_patchAddressModelDomain : modelSquare ⊆ patchAddressModelDomain := by |
| intro p hp |
| have hnorm : ‖p‖ ≤ 3 := by |
| apply (pi_norm_le_iff_of_nonneg (by norm_num : (0 : ℝ) ≤ 3)).mpr |
| intro i |
| simpa only [Real.norm_eq_abs] using (abs_le.mpr ⟨hp.1 i,hp.2 i⟩) |
| change dist p 0 < 13/4 |
| rw [dist_zero_right] |
| linarith |
|
|
| theorem patchAddressModelDomain_mapsTo_square (a : PatchAddress) : |
| MapsTo (affineCoordinates (patchAddressCenter a) (patchAddressMatrix a)) patchAddressModelDomain square := by |
| intro q hq |
| apply patch_core_image_mem_square a.1 a.2.1 a.2.2.1 a.2.2.2.val a.2.2.2.property |
| exact (by simpa only [patchAddressModelDomain,Metric.mem_ball,dist_zero_right] using hq : ‖q‖ < 13/4).le |
| |
| theorem patchAddressInitialMetric_smoothPositive (a : PatchAddress) : |
| SmoothPositiveOn (patchAddressInitialMetric a) patchAddressModelDomain := by |
| have h := affinePullbackMetric_smoothPositive initialAssemblyMetric_smoothPositiveOn |
| (patchAddressCenter a) (patchAddressMatrix a) (Matrix.mulVec_injective_of_isUnit (patchAddressMatrix_isUnit a)) |
| exact ⟨fun i j => (h.1 i j).mono (patchAddressModelDomain_mapsTo_square a), |
| fun p hp => h.2 p (patchAddressModelDomain_mapsTo_square a hp)⟩ |
| |
| theorem patchAddressInitialMetric_curvature (a : PatchAddress) {p : Coord} |
| (hp : p ∈ patchAddressModelDomain) : |
| gaussianCurvature (patchAddressInitialMetric a) p = modelCurvature (patchAddressAmplitude a) p := by |
| apply initialAssemblyMetric_model_in_patch_coordinates a.1 a.2.1 a.2.2.1 a.2.2.2.val a.2.2.2.property |
| exact (by simpa only [patchAddressModelDomain,Metric.mem_ball,dist_zero_right] using hp : ‖p‖ < 13/4).le |
| |
| end SmoothLocal.Geometry |
| |
| end |
| |
| end OAI |
| |
