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695bd70 | 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | 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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