File size: 3,942 Bytes
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 | 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
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