| import Mathlib |
| import OAI.Geometry.SmoothYau.Estimates.LpInverseGramFstContraction |
| import OAI.Geometry.SmoothYau.Model |
| import OAI.Geometry.SmoothYau.Smoothness.LinearPowerDirectional |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace YauCounterexamples |
| section |
| open Set Filter Function Manifold Module |
| open scoped Topology ContDiff InnerProductSpace |
| variable {A : Type*} [NormedAddCommGroup A] [InnerProductSpace ℝ A] |
| [FiniteDimensional ℝ A] {n : ℕ} [Fact (Module.finrank ℝ A = n+1)] |
| omit [FiniteDimensional ℝ A] |
| def unitSphereFrame (p : Metric.sphere (0 : A) 1) : Euclidean n →ₗᵢ[ℝ] A := |
| (ℝ ∙ (-(p : A)))ᗮ.subtypeₗᵢ.comp |
| (OrthonormalBasis.fromOrthogonalSpanSingleton n (ne_zero_of_mem_unit_sphere (-p))).repr.symm.toLinearIsometry |
|
|
| lemma unitSphereFrame_orthogonal (p : Metric.sphere (0 : A) 1) (w : Euclidean n) : |
| inner ℝ (p : A) (unitSphereFrame p w) = 0 := by |
| have hh := ((OrthonormalBasis.fromOrthogonalSpanSingleton n |
| (ne_zero_of_mem_unit_sphere (-p))).repr.symm w).property |
| rw [Submodule.mem_orthogonal_singleton_iff_inner_left] at hh |
| change inner ℝ (p : A) _ = 0 |
| rw [real_inner_comm] |
| change inner ℝ _ (-(p : A)) = 0 at hh |
| rw [inner_neg_right,neg_eq_zero] at hh |
| exact hh |
|
|
| lemma unitSphere_chart_symm (p : Metric.sphere (0 : A) 1) : |
| (fun y => ((chartAt (Euclidean n) p).symm y : A)) = |
| roundChart (p : A) (unitSphereFrame p) := by |
| funext y |
| change ((stereographic' n (-p)).symm y : A) = _ |
| rw [stereographic'_symm_apply] |
| simp only [roundChart,unitSphereFrame,stereoInvFunAux,smul_add,smul_smul] |
| rfl |
|
|
| lemma unitSphere_chart_center (p : Metric.sphere (0 : A) 1) : chartAt (Euclidean n) p p = 0 := by |
| have hh : (chartAt (Euclidean n) p).symm 0 = p := by |
| apply Subtype.ext |
| change (fun y => ((chartAt (Euclidean n) p).symm y : A)) 0 = _ |
| rw [unitSphere_chart_symm,roundChart_zero] |
| have hz : (0 : Euclidean n) ∈ (chartAt (Euclidean n) p).target := by |
| change 0 ∈ (stereographic' n (-p)).target |
| simp |
| exact (congrArg (chartAt (Euclidean n) p) hh).symm.trans ((chartAt (Euclidean n) p).right_inv hz) |
|
|
| lemma unitSphere_chart_target (p : Metric.sphere (0 : A) 1) : (chartAt (Euclidean n) p).target = univ := by |
| change (stereographic' n (-p)).target = _ |
| simp |
|
|
| lemma unitSphere_coordinate_embedding (p : Metric.sphere (0 : A) 1) (y : Euclidean n) (v : Euclidean n) : |
| mfderiv 𝓘(ℝ,Euclidean n) 𝓘(ℝ,A) |
| (fun q : Metric.sphere (0 : A) 1 => (q : A)) ((chartAt (Euclidean n) p).symm y) |
| (mfderiv 𝓘(ℝ,Euclidean n) 𝓘(ℝ,Euclidean n) (chartAt (Euclidean n) p).symm y v) = |
| fderiv ℝ (roundChart (p : A) (unitSphereFrame p)) y v := by |
| have hy : y ∈ (chartAt (Euclidean n) p).target := by rw [unitSphere_chart_target]; trivial |
| have hh := mfderiv_comp y |
| ((contMDiff_coe_sphere (m := ∞)).mdifferentiable (by simp)).mdifferentiableAt |
| ((contMDiffAt_symm_of_mem_maximalAtlas (IsManifold.chart_mem_maximalAtlas (n := ∞) p) hy).mdifferentiableAt (by simp)) |
| have he := congrArg (fun L => L v) hh |
| rw [mfderiv_eq_fderiv] at he |
| rw [← unitSphere_chart_symm] |
| exact he.symm |
|
|
| end |
|
|
|
|
| open Set Filter Function Manifold Module |
| open scoped Topology ContDiff InnerProductSpace |
| variable {A : Type*} [NormedAddCommGroup A] [InnerProductSpace ℝ A] |
| [FiniteDimensional ℝ A] {n : ℕ} [Fact (Module.finrank ℝ A = n+1)] |
| omit [FiniteDimensional ℝ A] |
| def unitSphereProjection (p : Metric.sphere (0 : A) 1) (a : A) : A := |
| a - inner ℝ (p : A) a • (p : A) |
|
|
| lemma unitSphereProjection_orthogonal (p : Metric.sphere (0 : A) 1) (a : A) : |
| unitSphereProjection p a ∈ (ℝ ∙ (-(p : A)))ᗮ := by |
| rw [Submodule.mem_orthogonal_singleton_iff_inner_left] |
| simp [unitSphereProjection,inner_sub_left,inner_smul_left,real_inner_comm a (p : A)] |
|
|
| def unitSphereCoordinates (p : Metric.sphere (0 : A) 1) (a : A) : Euclidean n := |
| (OrthonormalBasis.fromOrthogonalSpanSingleton n (ne_zero_of_mem_unit_sphere (-p))).repr |
| ⟨unitSphereProjection p a,unitSphereProjection_orthogonal p a⟩ |
|
|
| lemma unitSphereFrame_coordinates (p : Metric.sphere (0 : A) 1) (a : A) : |
| unitSphereFrame (n:=n) p (unitSphereCoordinates (n:=n) p a) = unitSphereProjection p a := by |
| change (((OrthonormalBasis.fromOrthogonalSpanSingleton n |
| (ne_zero_of_mem_unit_sphere (-p))).repr.symm |
| ((OrthonormalBasis.fromOrthogonalSpanSingleton n |
| (ne_zero_of_mem_unit_sphere (-p))).repr ⟨unitSphereProjection p a,_⟩)) : A) = _ |
| rw [LinearIsometryEquiv.symm_apply_apply] |
|
|
| lemma unitSphereCoordinates_inner (p : Metric.sphere (0 : A) 1) (a b : A) : |
| inner ℝ (unitSphereCoordinates (n:=n) p a) (unitSphereCoordinates (n:=n) p b) = |
| inner ℝ a b - inner ℝ (p : A) a * inner ℝ (p : A) b := by |
| rw [← (unitSphereFrame (n:=n) p).inner_map_map,unitSphereFrame_coordinates,unitSphereFrame_coordinates] |
| simp [unitSphereProjection,inner_sub_left,inner_sub_right,inner_smul_left,inner_smul_right, |
| real_inner_comm a (p : A)] |
|
|
| lemma unitSphereFrame_inner (p : Metric.sphere (0 : A) 1) (a : A) (w : Euclidean n) : |
| inner ℝ (unitSphereFrame (n:=n) p w) a = inner ℝ w (unitSphereCoordinates (n:=n) p a) := by |
| rw [← (unitSphereFrame (n:=n) p).inner_map_map,unitSphereFrame_coordinates] |
| have hh := unitSphereFrame_orthogonal p w |
| rw [real_inner_comm] at hh |
| simp [unitSphereProjection,inner_sub_right,inner_smul_right,hh] |
|
|
| lemma unitSphere_inverse_gram_contraction (p : Metric.sphere (0 : A) 1) (a b : A) : |
| ∑ i, ∑ j, (Matrix.gram ℝ (Module.finBasis ℝ (Euclidean n)))⁻¹ i j * |
| inner ℝ (unitSphereFrame (n:=n) p (Module.finBasis ℝ (Euclidean n) j)) a * |
| inner ℝ (unitSphereFrame (n:=n) p (Module.finBasis ℝ (Euclidean n) i)) b = |
| inner ℝ a b - inner ℝ (p : A) a * inner ℝ (p : A) b := by |
| simp_rw [unitSphereFrame_inner] |
| rw [inverse_gram_contraction,unitSphereCoordinates_inner] |
|
|
|
|
| end YauCounterexamples |
| end |
|
|
| end OAI |
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|
