| import OAI.Geometry.IsometricImmersion.Metrics.PatchTensorRestriction | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Function | |
| open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.Geometry | |
| open SmoothLocal.Perturbation | |
| theorem patchAddress_inverse_mulVec_injective (a : PatchAddress) : | |
| Injective ((patchAddressMatrix a)⁻¹).mulVec := by | |
| intro v w hvw | |
| have h := congrArg (fun u : Coord => patchAddressMatrix a *ᵥ u) hvw | |
| have hR := Matrix.mul_nonsing_inv (patchAddressMatrix a) | |
| ((Matrix.isUnit_iff_isUnit_det _).mp (patchAddressMatrix_isUnit a)) | |
| simpa only [Matrix.mulVec_mulVec,hR,Matrix.one_mulVec] using h | |
| theorem patchAddress_local_inverse_congruence (g : MetricField) | |
| (eta : PatchAddress → SymmetricPerturbation) (a : PatchAddress) (p : Coord) : | |
| (patchAddressMatrix a)⁻¹ᵀ * | |
| localPatchMetric g eta a (affineInverseCoordinates (patchAddressCenter a) (patchAddressMatrix a) p) * | |
| (patchAddressMatrix a)⁻¹ = (g + patchAddressTensor eta a) p := by | |
| have h := congrFun (affinePushforward_pullback g (patchAddressCenter a) | |
| (patchAddressMatrix a) (patchAddressMatrix_isUnit a)) p | |
| change (patchAddressMatrix a)⁻¹ᵀ * | |
| affinePullbackMetric g (patchAddressCenter a) (patchAddressMatrix a) | |
| (affineInverseCoordinates (patchAddressCenter a) (patchAddressMatrix a) p) * | |
| (patchAddressMatrix a)⁻¹ = g p at h | |
| simp only [localPatchMetric,Pi.add_apply,Matrix.mul_add,Matrix.add_mul,h] | |
| rfl | |
| theorem patchAddress_add_posDef_of_local_metric | |
| {g : MetricField} {U : Set Coord} (hg : ∀ p ∈ U, (g p).PosDef) | |
| (eta : PatchAddress → SymmetricPerturbation) (a : PatchAddress) | |
| (hlocal : ∀ q ∈ modelSquare, (localPatchMetric g eta a q).PosDef) | |
| {p : Coord} (hp : p ∈ U) : ((g + patchAddressTensor eta a) p).PosDef := by | |
| let q := affineInverseCoordinates (patchAddressCenter a) (patchAddressMatrix a) p | |
| by_cases hq : q ∈ patchBox | |
| · have hpos := (hlocal q (patchBox_subset_modelSquare hq)).conjTranspose_mul_mul_same | |
| (patchAddress_inverse_mulVec_injective a) | |
| simpa only [q, Matrix.conjTranspose_eq_transpose_of_trivial, | |
| patchAddress_local_inverse_congruence] using hpos | |
| · have hz : perturbationTensor (eta a) q = 0 := perturbationTensor_zero_off_patch (eta a) hq | |
| have hpzero : patchAddressTensor eta a p = 0 := by | |
| change (patchAddressMatrix a)⁻¹ᵀ * perturbationTensor (eta a) q * (patchAddressMatrix a)⁻¹ = 0 | |
| rw [hz,Matrix.mul_zero,Matrix.zero_mul] | |
| change (g p + patchAddressTensor eta a p).PosDef | |
| rw [hpzero,add_zero] | |
| exact hg p hp | |
| theorem assembledPatchMetric_smoothPositive_of_local_metrics {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, ∀ q ∈ modelSquare, (localPatchMetric g eta a q).PosDef) : | |
| SmoothPositiveOn (assembledPatchMetric g eta) U := | |
| assembledPatchMetric_smoothPositive he hg eta hsmall | |
| (fun a _ hp => patchAddress_add_posDef_of_local_metric hg.2 eta a (hlocal a) hp.1) | |
| theorem assembledPatchMetric_curvature_on_disk_of_local_metrics {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, ∀ q ∈ modelSquare, (localPatchMetric g eta a q).PosDef) | |
| (n : ℕ) {p : Coord} (hp : p ∈ accumulatingDisk n) (hpU : p ∈ U) : | |
| gaussianCurvature (assembledPatchMetric g eta) p = gaussianCurvature g p := | |
| assembledPatchMetric_curvature_on_disk he hg hU eta hsmall | |
| (fun a _ hp => patchAddress_add_posDef_of_local_metric hg.2 eta a (hlocal a) hp.1) n hp hpU | |
| end SmoothLocal.Geometry | |
| end | |
| end OAI | |