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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 | 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
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