import Mathlib import OAI.Geometry.SmoothYau.Smoothness.ChartTensorCoordLocal1 namespace OAI noncomputable section namespace YauCounterexamples section open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter Metric open scoped Topology InnerProductSpace open Set Filter Function Metric open scoped Topology open Set Filter Function Metric open scoped Topology open Set Filter Manifold open scoped Topology ContDiff Matrix.Norms.Elementwise variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] local instance mcfDualNorm : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance local instance mcfDualSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance local instance mcfFormNorm : NormedAddCommGroup (CoordinateForm E) := inferInstance local instance mcfFormSpace : NormedSpace ℝ (CoordinateForm E) := inferInstance private local instance mcfMatrixMapNorm {ι κ : Type*} [Fintype ι] [Fintype κ] : NormedAddCommGroup (Matrix ι κ ℝ →L[ℝ] CoordinateForm (E)) := inferInstanceAs (NormedAddCommGroup ((ι → κ → ℝ) →L[ℝ] CoordinateForm (E))) private local instance mcfMapMatrixNorm {ι κ : Type*} [Fintype ι] [Fintype κ] : NormedAddCommGroup (E →L[ℝ] Matrix ι κ ℝ) := inferInstanceAs (NormedAddCommGroup (E →L[ℝ] (ι → κ → ℝ))) def metricMatrixFormCLM : Matrix (CoordIndex E) (CoordIndex E) ℝ →L[ℝ] CoordinateForm E := ({ toFun := metricMatrixForm map_add' := by intro A B simp only [metricMatrixForm, Matrix.add_apply, add_smul, Finset.sum_add_distrib] map_smul' := by intro c A simp only [metricMatrixForm, Matrix.smul_apply, smul_eq_mul, mul_smul, Finset.smul_sum, RingHom.id_apply] } : Matrix (CoordIndex E) (CoordIndex E) ℝ →ₗ[ℝ] CoordinateForm E).toContinuousLinearMap lemma metricMatrixFormCLM_apply (A : Matrix (CoordIndex E) (CoordIndex E) ℝ) : metricMatrixFormCLM A = (metricMatrixForm A : CoordinateForm E) := rfl omit [FiniteDimensional ℝ E] in lemma matrix_fderiv_norm_le {ι κ : Type*} [Fintype ι] [Fintype κ] {A : E → Matrix ι κ ℝ} {x : E} (hA : DifferentiableAt ℝ A x) {ε : ℝ} (hε : 0 ≤ ε) (hb : ∀ i j, ‖fderiv ℝ (fun y => A y i j) x‖ ≤ ε) : ‖fderiv ℝ A x‖ ≤ ε := by apply ContinuousLinearMap.opNorm_le_bound _ hε intro v apply (pi_norm_le_iff_of_nonneg (mul_nonneg hε (norm_nonneg v))).mpr intro i apply (pi_norm_le_iff_of_nonneg (mul_nonneg hε (norm_nonneg v))).mpr intro j have he : fderiv ℝ (fun y => A y i j) x v = (fderiv ℝ A x v) i j := by rw [fderiv_apply (differentiableAt_pi.mp hA i) j, fderiv_apply hA i] rfl erw [← he] exact (ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_right (hb i j) (norm_nonneg v)) theorem metricMatrixForm_first_jet_bound {A B : E → Matrix (CoordIndex E) (CoordIndex E) ℝ} {x : E} (hA : DifferentiableAt ℝ A x) (hB : DifferentiableAt ℝ B x) {ε : ℝ} (hε : 0 ≤ ε) (h₀ : ∀ i j, |A x i j-B x i j| ≤ ε) (h₁ : ∀ i j, ‖fderiv ℝ (fun y => A y i j-B y i j) x‖ ≤ ε) : ‖metricMatrixForm (A x)-metricMatrixForm (B x)‖ ≤ ‖metricMatrixFormCLM (E := E)‖*ε ∧ ‖fderiv ℝ (fun y => metricMatrixForm (A y)) x- fderiv ℝ (fun y => metricMatrixForm (B y)) x‖ ≤ ‖metricMatrixFormCLM (E := E)‖*ε := by let L := metricMatrixFormCLM (E := E) have hF : DifferentiableAt ℝ (fun y => A y-B y) x := hA.sub hB have hn : ‖A x-B x‖ ≤ ε := by apply (pi_norm_le_iff_of_nonneg hε).mpr intro i apply (pi_norm_le_iff_of_nonneg hε).mpr intro j exact h₀ i j have hDn : ‖fderiv ℝ (fun y => A y-B y) x‖ ≤ ε := matrix_fderiv_norm_le hF hε h₁ constructor · change ‖L (A x)-L (B x)‖ ≤ ‖L‖*ε rw [← map_sub] exact (L.le_opNorm _).trans (mul_le_mul_of_nonneg_left hn (norm_nonneg L)) · have hLA : DifferentiableAt ℝ (fun y => metricMatrixForm (A y)) x := L.differentiableAt.comp x hA have hLB : DifferentiableAt ℝ (fun y => metricMatrixForm (B y)) x := L.differentiableAt.comp x hB rw [← fderiv_fun_sub hLA hLB] have he : (fun y => metricMatrixForm (A y)-metricMatrixForm (B y)) = fun y => L (A y-B y) := by funext y exact (map_sub L _ _).symm rw [he] have hd : fderiv ℝ (fun y => L (A y-B y)) x = L.comp (fderiv ℝ (fun y => A y-B y) x) := (L.hasFDerivAt.comp x hF.hasFDerivAt).fderiv rw [hd] exact (L.opNorm_comp_le _).trans (mul_le_mul_of_nonneg_left hDn (norm_nonneg L)) end section open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter Metric open scoped Topology InnerProductSpace open Set Filter Function Metric open scoped Topology open Set Filter Function Metric open scoped Topology open Set Filter Metric open scoped Topology InnerProductSpace variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] local instance hmsDualNorm : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance local instance hmsDualSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance local instance hmsFormNorm : NormedAddCommGroup (CoordinateForm E) := inferInstance local instance hmsFormSpace : NormedSpace ℝ (CoordinateForm E) := inferInstance local instance hmsDerivNorm : NormedAddCommGroup (E →L[ℝ] CoordinateForm E) := inferInstance local instance hmsDerivSpace : NormedSpace ℝ (E →L[ℝ] CoordinateForm E) := inferInstance abbrev HessianJetData (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] := (CoordinateForm E × (E →L[ℝ] CoordinateForm E)) × ((E →L[ℝ] ℝ) × CoordinateForm E) local instance hmsJetNorm : NormedAddCommGroup (HessianJetData E) := inferInstance local instance hmsJetSpace : NormedSpace ℝ (HessianJetData E) := inferInstance def christoffelFromJet (B : CoordinateForm E) (D : E →L[ℝ] CoordinateForm E) : E →L[ℝ] E →L[ℝ] E := ((ContinuousLinearMap.compL ℝ E (E →L[ℝ] ℝ) E) B.inverse).comp (coordinateKoszul D) def metricHessianFromFullJet (z : HessianJetData E) : CoordinateForm E := z.2.2 - (ContinuousLinearMap.compL ℝ E E ℝ z.2.1).comp (christoffelFromJet z.1.1 z.1.2) lemma continuous_coordinateKoszul : Continuous (coordinateKoszul : (E →L[ℝ] CoordinateForm E) → E →L[ℝ] CoordinateForm E) := by have hDf : Continuous (fun D : E →L[ℝ] CoordinateForm E => D.flip) := (ContinuousLinearMap.flipₗᵢ ℝ E E (E →L[ℝ] ℝ)).toContinuousLinearMap.continuous exact ((continuous_id.add hDf).sub (continuous_const.clm_comp hDf)).const_smul (1/2 : ℝ) lemma continuousAt_metricHessianFromFullJet {z : HessianJetData E} (hB : z.1.1.IsInvertible) : ContinuousAt (metricHessianFromFullJet : HessianJetData E → CoordinateForm E) z := by have hp : ContinuousAt (fun w : HessianJetData E => w.1.1) z := continuous_fst.fst.continuousAt have hi : ContinuousAt (fun w : HessianJetData E => w.1.1.inverse) z := (hB.contDiffAt_map_inverse (n := ∞)).continuousAt.comp (f := fun w : HessianJetData E => w.1.1) (x := z) hp have hK : Continuous (fun z : HessianJetData E => coordinateKoszul z.1.2) := continuous_coordinateKoszul.comp continuous_fst.snd have hΓ : ContinuousAt (fun z : HessianJetData E => christoffelFromJet z.1.1 z.1.2) z := (continuousAt_const.clm_apply hi).clm_comp hK.continuousAt exact continuous_snd.snd.continuousAt.sub ((continuousAt_const.clm_apply continuous_snd.fst.continuousAt).clm_comp hΓ) def metricScalarJet (g : SmoothMetric E E) (f : E → ℝ) (x : E) : HessianJetData E := ((selfMetricFlat g x,fderiv ℝ (selfMetricFlat g) x), (fderiv ℝ f x,fderiv ℝ (fderiv ℝ f) x)) lemma metricHessianFromFullJet_metricScalarJet (g : SmoothMetric E E) (f : E → ℝ) (x : E) : metricHessianFromFullJet (metricScalarJet g f x) = actualCoordinateHessian g f x := rfl omit [FiniteDimensional ℝ E] in lemma continuous_metricScalarJet (g : SmoothMetric E E) {f : E → ℝ} (hf : ContDiff ℝ ∞ f) : Continuous (metricScalarJet g f) := by exact ((contDiff_selfMetricFlat g).continuous.prodMk ((contDiff_selfMetricFlat g).continuous_fderiv (by simp))).prodMk ((hf.continuous_fderiv (by simp)).prodMk ((hf.fderiv_right (m := ∞) (by simp)).continuous_fderiv (by simp))) def hessianJetMarginSet : Set (HessianJetData E) := {z | hessianMargins z.1.1 (metricHessianFromFullJet z)} lemma metricScalarJet_margin_interior (g : SmoothMetric E E) (f : E → ℝ) (x : E) (h : hessianMargins (selfMetricFlat g x) (actualCoordinateHessian g f x)) : metricScalarJet g f x ∈ interior (hessianJetMarginSet (E := E)) := by apply mem_interior_iff_mem_nhds.mpr have hc : ContinuousAt (fun z : HessianJetData E => (z.1.1,metricHessianFromFullJet z)) (metricScalarJet g f x) := continuous_fst.fst.continuousAt.prodMk (continuousAt_metricHessianFromFullJet (selfMetricFlat_invertible g x)) exact hc ((isOpen_hessianMargins (E := E)).mem_nhds h) omit [FiniteDimensional ℝ E] in lemma dist_metricScalarJet_lt (g g₀ : SmoothMetric E E) (f : E → ℝ) (x : E) {δ : ℝ} (hδ : 0 < δ) (h₀ : ‖selfMetricFlat g x-selfMetricFlat g₀ x‖ < δ) (h₁ : ‖fderiv ℝ (selfMetricFlat g) x-fderiv ℝ (selfMetricFlat g₀) x‖ < δ) : dist (metricScalarJet g f x) (metricScalarJet g₀ f x) < δ := by change max (max (dist (selfMetricFlat g x) (selfMetricFlat g₀ x)) (dist (fderiv ℝ (selfMetricFlat g) x) (fderiv ℝ (selfMetricFlat g₀) x))) (dist (fderiv ℝ f x,fderiv ℝ (fderiv ℝ f) x) (fderiv ℝ f x,fderiv ℝ (fderiv ℝ f) x)) < δ rw [dist_self, dist_eq_norm, dist_eq_norm] exact max_lt (max_lt h₀ h₁) hδ lemma compact_map_uniform_interior {X Y : Type*} [TopologicalSpace X] [MetricSpace Y] {K : Set X} (hK : IsCompact K) {F : X → Y} (hF : Continuous F) {U : Set Y} (hU : ∀ x ∈ K, F x ∈ interior U) : ∃ δ > 0, ∀ x ∈ K, ∀ y, dist y (F x) < δ → y ∈ U := by have hi : F '' K ⊆ interior U := by rintro _ ⟨x,hx,rfl⟩ exact hU x hx obtain ⟨δ,hδ,hδU⟩ := (hK.image hF).exists_thickening_subset_open isOpen_interior hi refine ⟨δ,hδ,?_⟩ intro x hx y hy exact interior_subset (hδU (mem_thickening_iff.mpr ⟨F x,⟨x,hx,rfl⟩,hy⟩)) theorem compact_hessianMargins_stability (g₀ : SmoothMetric E E) {f : E → ℝ} (hf : ContDiff ℝ ∞ f) {K : Set E} (hK : IsCompact K) (hmargin : ∀ x ∈ K, hessianMargins (selfMetricFlat g₀ x) (actualCoordinateHessian g₀ f x)) : ∃ δ > 0, ∀ g : SmoothMetric E E, (∀ x ∈ K, ‖selfMetricFlat g x-selfMetricFlat g₀ x‖ < δ ∧ ‖fderiv ℝ (selfMetricFlat g) x-fderiv ℝ (selfMetricFlat g₀) x‖ < δ) → ∀ x ∈ K, hessianMargins (selfMetricFlat g x) (actualCoordinateHessian g f x) := by obtain ⟨δ,hδ,hstable⟩ := compact_map_uniform_interior hK (continuous_metricScalarJet g₀ hf) (fun x hx => metricScalarJet_margin_interior g₀ f x (hmargin x hx)) refine ⟨δ,hδ,?_⟩ intro g hg x hx exact hstable x hx (metricScalarJet g f x) (dist_metricScalarJet_lt g g₀ f x hδ (hg x hx).1 (hg x hx).2) end section open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter Metric open scoped Topology InnerProductSpace open Set Filter Function Metric open scoped Topology open Set Filter Function Metric open scoped Topology open Set Filter Manifold open scoped Topology ContDiff Matrix.Norms.Elementwise variable {n : ℕ} local instance spnENorm : NormedAddCommGroup (Euclidean n) := inferInstance local instance spnESpace : NormedSpace ℝ (Euclidean n) := inferInstance local instance spnDualNorm : NormedAddCommGroup (Euclidean n →L[ℝ] ℝ) := inferInstance local instance spnDualSpace : NormedSpace ℝ (Euclidean n →L[ℝ] ℝ) := inferInstance local instance spnFormNorm : NormedAddCommGroup (CoordinateForm (Euclidean n)) := inferInstance local instance spnFormSpace : NormedSpace ℝ (CoordinateForm (Euclidean n)) := inferInstance local instance spnDerivNorm : NormedAddCommGroup (Euclidean n →L[ℝ] CoordinateForm (Euclidean n)) := inferInstance local instance spnDerivSpace : NormedSpace ℝ (Euclidean n →L[ℝ] CoordinateForm (Euclidean n)) := inferInstance private local instance spnMatrixMapNorm {ι κ : Type*} [Fintype ι] [Fintype κ] : NormedAddCommGroup (Matrix ι κ ℝ →L[ℝ] CoordinateForm (Euclidean n)) := inferInstanceAs (NormedAddCommGroup ((ι → κ → ℝ) →L[ℝ] CoordinateForm (Euclidean n))) lemma sphere_metricCoefficients_smooth (g : SmoothMetric (Euclidean n) (Sphere n)) (p : Sphere n) : ContDiff ℝ ∞ (metricCoefficients g p) := by apply contDiff_pi.mpr intro i apply contDiff_pi.mpr intro j exact contDiffOn_univ.mp (by simpa only [sphere_chart_target] using contDiffOn_metricCoefficient g p i j) theorem sphere_metric_first_jet_neighborhood (g₀ : SmoothMetric (Euclidean n) (Sphere n)) (p : Sphere n) {K : Set (Euclidean n)} (hK : IsCompact K) {δ : ℝ} (hδ : 0 < δ) : IsSmoothNeighborhood g₀ {g | ∀ x ∈ K, ‖selfMetricFlat (sphereChartMetric g p) x-selfMetricFlat (sphereChartMetric g₀ p) x‖ < δ ∧ ‖fderiv ℝ (selfMetricFlat (sphereChartMetric g p)) x- fderiv ℝ (selfMetricFlat (sphereChartMetric g₀ p)) x‖ < δ} := by let L := metricMatrixFormCLM (E := Euclidean n) let ε := δ/(‖L‖+1) have hden : 0 < ‖L‖+1 := by positivity have hε : 0 < ε := div_pos hδ hden have hmul : ‖L‖*ε < δ := by calc ‖L‖*ε < (‖L‖+1)*ε := mul_lt_mul_of_pos_right (by linarith) hε _ = δ := by dsimp [ε]; field_simp let test : Fin 2 × CoordIndex (Euclidean n) × CoordIndex (Euclidean n) → CoordinateTest (Euclidean n) (Sphere n) := fun q => { center := p, compactSet := K, isCompact := hK, inTarget := by rw [sphere_chart_target]; exact subset_univ K, order := q.1.val, row := q.2.1, column := q.2.2 } let tests := (Finset.univ : Finset (Fin 2 × CoordIndex (Euclidean n) × CoordIndex (Euclidean n))).toList.map test refine ⟨tests,ε,hε,?_⟩ intro g hg x hx have htest (k : Fin 2) (i j : CoordIndex (Euclidean n)) : test (k,i,j) ∈ tests := by exact List.mem_map.mpr ⟨(k,i,j),by simp,rfl⟩ have h₀ (i j : CoordIndex (Euclidean n)) : |metricCoefficients g p x i j-metricCoefficients g₀ p x i j| ≤ ε := by have hh := hg (test (0,i,j)) (htest 0 i j) x hx simpa only [test,Fin.val_zero,norm_iteratedFDeriv_zero,Real.norm_eq_abs] using hh.le have h₁ (i j : CoordIndex (Euclidean n)) : ‖fderiv ℝ (fun y => metricCoefficients g p y i j-metricCoefficients g₀ p y i j) x‖ ≤ ε := by have hh := hg (test (1,i,j)) (htest 1 i j) x hx simpa only [test,Fin.val_one,norm_iteratedFDeriv_one] using hh.le have hb := metricMatrixForm_first_jet_bound ((sphere_metricCoefficients_smooth g p).differentiable (by simp) x) ((sphere_metricCoefficients_smooth g₀ p).differentiable (by simp) x) hε.le h₀ h₁ have he (g : SmoothMetric (Euclidean n) (Sphere n)) : selfMetricFlat (sphereChartMetric g p) = fun y => metricMatrixForm (metricCoefficients g p y) := funext (sphereChartMetric_selfFlat g p) rw [he g,he g₀] exact ⟨hb.1.trans_lt hmul,hb.2.trans_lt hmul⟩ theorem sphericalBaseProfile_neighborhood (g₀ : SmoothMetric (Euclidean 3) (Sphere 3)) (hg₀ : IsRound g₀) (p : Sphere 3) (a b : Euclidean 4) (ha : inner ℝ a a = 1) (hb : inner ℝ b b = 1) (hab : inner ℝ a b = 0) (t : ℝ) {K : Set (Euclidean 3)} (hK : IsCompact K) (hr : ∀ y ∈ K, (inner ℝ (roundChart (p : Euclidean 4) (sphereFrame p) y) a)^2 + (inner ℝ (roundChart (p : Euclidean 4) (sphereFrame p) y) b)^2 ≤ 1/8) : IsSmoothNeighborhood g₀ {g | ∀ y ∈ K, hessianMargins (selfMetricFlat (sphereChartMetric g p) y) (actualCoordinateHessian (sphereChartMetric g p) (sphericalBaseProfile a b t ∘ (chartAt (Euclidean 3) p).symm) y)} := by have hm (y : Euclidean 3) (hy : y ∈ K) : hessianMargins (selfMetricFlat (sphereChartMetric g₀ p) y) (actualCoordinateHessian (sphereChartMetric g₀ p) (sphericalBaseProfile a b t ∘ (chartAt (Euclidean 3) p).symm) y) := by have hh := sphericalBaseProfile_chart_margins g₀ hg₀ p a b ha hb hab t y (hr y hy) exact hessianMargins_of_strong _ _ (fun v hv => by rw [selfMetricFlat_apply]; exact (sphereChartMetric g₀ p).pos _ _ hv) hh.1 hh.2 obtain ⟨δ,hδ,hstable⟩ := compact_hessianMargins_stability (sphereChartMetric g₀ p) (sphericalBaseProfile_chart_smooth p a b t) hK hm obtain ⟨tests,ε,hε,htests⟩ := sphere_metric_first_jet_neighborhood g₀ p hK hδ refine ⟨tests,ε,hε,?_⟩ intro g hg exact hstable (sphereChartMetric g p) (htests g hg) end section open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter Metric open scoped Topology InnerProductSpace open Set Filter Function Metric open scoped Topology open Set Filter Function Metric open scoped Topology open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter Metric open scoped Topology InnerProductSpace open Set Filter Function Metric open scoped Topology open Set Filter Function Metric open scoped Topology open Set Filter Manifold Metric open scoped Topology ContDiff lemma IsSmoothNeighborhood.mono {E M : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] (g₀ : SmoothMetric E M) {N O : Set (SmoothMetric E M)} (hN : IsSmoothNeighborhood g₀ N) (hNO : N ⊆ O) : IsSmoothNeighborhood g₀ O := by obtain ⟨tests,ε,hε,h⟩ := hN exact ⟨tests,ε,hε,fun g hg => hNO (h g hg)⟩ lemma IsSmoothNeighborhood.inter {E M : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] (g₀ : SmoothMetric E M) {N O : Set (SmoothMetric E M)} (hN : IsSmoothNeighborhood g₀ N) (hO : IsSmoothNeighborhood g₀ O) : IsSmoothNeighborhood g₀ (N ∩ O) := by obtain ⟨t,ε,hε,h⟩ := hN obtain ⟨s,δ,hδ,k⟩ := hO refine ⟨t++s,min ε δ,lt_min hε hδ,fun g hg => ⟨h g ?_,k g ?_⟩⟩ · intro a ha y hy exact (hg a (List.mem_append_left _ ha) y hy).trans_le (min_le_left _ _) · intro a ha y hy exact (hg a (List.mem_append_right _ ha) y hy).trans_le (min_le_right _ _) lemma chartMetricForm_domination_transfer {E M : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] (g₀ g : SmoothMetric E M) (p : M) {x : M} (hx : x ∈ (chartAt E p).source) (a : ℝ) (h : ∀ v : E, chartMetricForm g₀ p (chartAt E p x) v v ≤ a*chartMetricForm g p (chartAt E p x) v v) : ∀ v : TangentSpace 𝓘(ℝ,E) x, g₀.inner x v v ≤ a*g.inner x v v := by let c := chartAt E p have hc : c.MDifferentiable 𝓘(ℝ,E) 𝓘(ℝ,E) := ⟨mdifferentiableOn_atlas (chart_mem_atlas E p),mdifferentiableOn_atlas_symm (chart_mem_atlas E p)⟩ have hh : ∀ v : TangentSpace 𝓘(ℝ,E) (c.symm (c x)), g₀.inner (c.symm (c x)) v v ≤ a*g.inner (c.symm (c x)) v v := by intro v obtain ⟨w,rfl⟩ := hc.symm.mfderiv_surjective (c.map_source hx) v have hw := h w erw [chartMetricForm_pairing, chartMetricForm_pairing] at hw exact hw have he : c.symm (c x) = x := c.left_inv hx exact he ▸ hh lemma small_bilinear_difference_domination {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (B G : E →L[ℝ] E →L[ℝ] ℝ) {m : ℝ} (hm : 0 < m) (hB : ∀ v, m*‖v‖^2 ≤ B v v) (hG : ‖G-B‖ < m/2) : ∀ v, B v v ≤ 4*G v v := by intro v have he : |G v v-B v v| ≤ (m/2)*‖v‖^2 := by calc _ = ‖(G-B) v v‖ := by simp _ ≤ ‖G-B‖*‖v‖*‖v‖ := (ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_right (ContinuousLinearMap.le_opNorm _ _) (norm_nonneg _)) _ ≤ _ := by nlinarith [mul_le_mul_of_nonneg_right hG.le (sq_nonneg ‖v‖)] have hl := hB v have hmnorm : 0 ≤ B v v := (mul_nonneg hm.le (sq_nonneg _)).trans hl have ha := (abs_le.mp he).1 nlinarith def sphericalHalf (p : Sphere 3) : Set (Sphere 3) := {q | 0 ≤ inner ℝ (p : Euclidean 4) (q : Euclidean 4)} lemma sphericalHalf_compact (p : Sphere 3) : IsCompact (sphericalHalf p) := (isClosed_le continuous_const (continuous_const.inner continuous_subtype_val)).isCompact lemma sphericalHalf_in_source (p : Sphere 3) : sphericalHalf p ⊆ (chartAt (Euclidean 3) p).source := by intro q hq rw [sphere_chart_source] change q ≠ -p intro he rw [he] at hq have hp : ‖(p : Euclidean 4)‖ = 1 := by simpa only [mem_sphere,dist_zero_right] using p.property change 0 ≤ inner ℝ (p : Euclidean 4) (-(p : Euclidean 4)) at hq rw [inner_neg_right,real_inner_self_eq_norm_sq,hp] at hq norm_num at hq lemma sphericalHalves_cover (p q : Sphere 3) : q ∈ sphericalHalf p ∨ q ∈ sphericalHalf (-p) := by change 0 ≤ inner ℝ (p : Euclidean 4) (q : Euclidean 4) ∨ 0 ≤ inner ℝ (-(p : Euclidean 4)) (q : Euclidean 4) rw [inner_neg_left] exact le_total 0 _ |>.imp id (fun h => neg_nonneg.mpr h) lemma sphericalHalf_domination_neighborhood (g₀ : SmoothMetric (Euclidean 3) (Sphere 3)) (p : Sphere 3) : IsSmoothNeighborhood g₀ {g | ∀ x ∈ sphericalHalf p, ∀ v : TangentSpace 𝓘(ℝ,Euclidean 3) x, g₀.inner x v v ≤ 4*g.inner x v v} := by let c := chartAt (Euclidean 3) p let K := c '' sphericalHalf p have hK : IsCompact K := (sphericalHalf_compact p).image_of_continuousOn (c.continuousOn.mono (sphericalHalf_in_source p)) obtain ⟨m,hm,hml⟩ := compact_selfMetricFlat_lower (sphereChartMetric g₀ p) hK apply IsSmoothNeighborhood.mono g₀ (sphere_metric_first_jet_neighborhood g₀ p hK (half_pos hm)) intro g hg x hx apply chartMetricForm_domination_transfer g₀ g p (sphericalHalf_in_source p hx) 4 have hxx : c x ∈ K := ⟨x,hx,rfl⟩ have hdom := small_bilinear_difference_domination _ _ hm (hml _ hxx) (hg _ hxx).1 simpa only [sphereChartMetric_selfFlat] using hdom theorem spherical_domination_neighborhood (g₀ : SmoothMetric (Euclidean 3) (Sphere 3)) : IsSmoothNeighborhood g₀ {g | ∀ x (v : TangentSpace 𝓘(ℝ,Euclidean 3) x), g₀.inner x v v ≤ 4*g.inner x v v} := by apply IsSmoothNeighborhood.mono g₀ (IsSmoothNeighborhood.inter g₀ (sphericalHalf_domination_neighborhood g₀ sourcePole) (sphericalHalf_domination_neighborhood g₀ (-sourcePole))) intro g hg x v rcases sphericalHalves_cover sourcePole x with h | h · exact hg.1 x h v · exact hg.2 x h v end section open Set Filter Manifold open scoped Topology ContDiff Matrix.Norms.Elementwise variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] lemma matrix_iteratedFDeriv_norm_le {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq ι] [DecidableEq κ] {A : E → Matrix ι κ ℝ} {x : E} (hA : ContDiffAt ℝ ∞ A x) (m : ℕ) {ε : ℝ} (hε : 0 ≤ ε) (hb : ∀ i j, ‖iteratedFDeriv ℝ m (fun y => A y i j) x‖ ≤ ε) : ‖iteratedFDeriv ℝ m A x‖ ≤ ε := by apply ContinuousMultilinearMap.opNorm_le_bound hε intro v have hp := Finset.prod_nonneg (fun k (_ : k ∈ Finset.univ) => norm_nonneg (v k)) apply (pi_norm_le_iff_of_nonneg (mul_nonneg hε hp)).mpr intro i apply (pi_norm_le_iff_of_nonneg (mul_nonneg hε hp)).mpr intro j let L : Matrix ι κ ℝ →L[ℝ] ℝ := (ContinuousLinearMap.proj j : (κ → ℝ) →L[ℝ] ℝ).comp (ContinuousLinearMap.proj i : Matrix ι κ ℝ →L[ℝ] (κ → ℝ)) have he : iteratedFDeriv ℝ m (fun y => A y i j) x v = (iteratedFDeriv ℝ m A x v) i j := congrArg (fun T => T v) (L.iteratedFDeriv_comp_left hA (ENat.natCast_le_of_coe_top_le_withTop le_rfl m)) rw [← he] exact (ContinuousMultilinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_right (hb i j) hp) variable [FiniteDimensional ℝ E] local instance matrixJetInst1 : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance local instance matrixJetInst2 : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance local instance matrixJetInst3 : NormedAddCommGroup (CoordinateForm E) := inferInstance local instance matrixJetInst4 : NormedSpace ℝ (CoordinateForm E) := inferInstance private local instance matrixJetMatrixMapNorm {ι κ : Type*} [Fintype ι] [Fintype κ] : NormedAddCommGroup (Matrix ι κ ℝ →L[ℝ] CoordinateForm (E)) := inferInstanceAs (NormedAddCommGroup ((ι → κ → ℝ) →L[ℝ] CoordinateForm (E))) lemma metricMatrixForm_iterated_jet_bound {A : E → Matrix (CoordIndex E) (CoordIndex E) ℝ} {x : E} (hA : ContDiffAt ℝ ∞ A x) (m : ℕ) {ε : ℝ} (hε : 0 ≤ ε) (hb : ∀ i j, ‖iteratedFDeriv ℝ m (fun y => A y i j) x‖ ≤ ε) : ‖iteratedFDeriv ℝ m (fun y => metricMatrixForm (A y)) x‖ ≤ ‖metricMatrixFormCLM (E:=E)‖*ε := by classical let L := metricMatrixFormCLM (E:=E) change ‖iteratedFDeriv ℝ m (L ∘ A) x‖ ≤ ‖L‖*ε erw [L.iteratedFDeriv_comp_left hA (ENat.natCast_le_of_coe_top_le_withTop le_rfl m)] exact (L.norm_compContinuousMultilinearMap_le _).trans (mul_le_mul_of_nonneg_left (matrix_iteratedFDeriv_norm_le hA m hε hb) (norm_nonneg L)) end open Set Filter Function Manifold Bundle TopologicalSpace open scoped Topology ContDiff Distributions BoundedContinuousFunction Matrix.Norms.Elementwise section CompactSmoothContinuity variable {P E F : Type*} [TopologicalSpace P] [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] lemma continuous_compactSmooth_of_jets (K : Compacts E) (f : P → 𝓓_{K}(E,F)) (hf : ∀ n, Continuous (fun z : P × E => iteratedFDeriv ℝ n (f z.1) z.2)) : Continuous f := by rw [continuous_iff_continuousAt] intro a₀ rw [ContinuousAt,(ContDiffMapSupportedIn.withSeminorms ℝ E F ⊤ K).tendsto_nhds] intro n ε hε have hj : Continuous (fun z : P × E => iteratedFDeriv ℝ n (f z.1-f a₀) z.2) := by have he : (fun z : P × E => iteratedFDeriv ℝ n (f z.1-f a₀) z.2) = (fun z : P × E => iteratedFDeriv ℝ n (f z.1) z.2-iteratedFDeriv ℝ n (f a₀) z.2) := by funext z exact iteratedFDeriv_sub_apply ((f z.1).contDiff.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl n)).contDiffAt ((f a₀).contDiff.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl n)).contDiffAt rw [he] exact (hf n).sub (((f a₀).contDiff.continuous_iteratedFDeriv (ENat.natCast_le_of_coe_top_le_withTop le_rfl n)).comp continuous_snd) have he : ∀ᶠ a in 𝓝 a₀, ∀ y ∈ (K : Set E), ‖iteratedFDeriv ℝ n (f a-f a₀) y‖ < ε/2 := by apply K.isCompact.eventually_forall_of_forall_eventually intro y hy exact hj.norm.continuousAt.eventually (gt_mem_nhds (by simpa using half_pos hε)) filter_upwards [he] with a ha apply lt_of_le_of_lt ?_ (half_lt_self hε) apply (ContDiffMapSupportedIn.seminorm_top_le_iff ℝ (half_pos hε).le _ _).mpr intro y hy exact (ha y hy).le end CompactSmoothContinuity namespace ContinuousSmoothFamilyOn variable {P E F ι : Type*} [TopologicalSpace P] [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [Fintype ι] lemma pi {f : P → E → ι → F} {U : Set E} (hf : ∀ i, ContinuousSmoothFamilyOn (fun p x => f p x i) U) : ContinuousSmoothFamilyOn f U := by classical have hh := sum Finset.univ (fun i p x => Pi.single i (f p x i)) (fun i _ => (hf i).postcomp (ContinuousLinearMap.single ℝ (fun _ : ι => F) i) (fun _ _ _ => (ContinuousLinearMap.single ℝ (fun _ : ι => F) i).contDiff.contDiffAt)) have he : (fun p x => ∑ i, Pi.single i (f p x i)) = f := by funext p x i simp rwa [he] at hh end ContinuousSmoothFamilyOn variable {P E M : Type*} [TopologicalSpace P] [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] [T2Space M] local instance compactSmoothContinuityInst1 : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance local instance compactSmoothContinuityInst2 : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance local instance compactSmoothContinuityInst3 : NormedAddCommGroup (MetricForm E) := inferInstanceAs (NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ)) local instance compactSmoothContinuityInst4 : NormedSpace ℝ (MetricForm E) := inferInstanceAs (NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ)) omit [T2Space M] in lemma continuous_chartTensorOfMetric (g₀ : SmoothMetric E M) (p : M) (K : Compacts E) (ht : (chartAt E p).target = univ) (q : P → SmoothMetric E M) (he : ∀ a x, x ∉ (chartAt E p).symm '' (K : Set E) → (q a).inner x=g₀.inner x) (hq : ∀ i j, ContinuousSmoothFamilyOn (fun a y => metricCoefficients (q a) p y i j) univ) : Continuous (fun a => chartTensorOfMetric g₀ p K ht (q a) (he a)) := by apply continuous_compactSmooth_of_jets have hf : ContinuousSmoothFamilyOn (fun a y => metricCoefficients (q a) p y) univ := ContinuousSmoothFamilyOn.pi (fun i => ContinuousSmoothFamilyOn.pi (hq i)) have hfm := hf.postcomp (metricMatrixFormCLM (E:=E)) (fun _ _ _ => (metricMatrixFormCLM (E:=E)).contDiff.contDiffAt) have hfb : ContinuousSmoothFamilyOn (fun (_ : P) y => chartMetricForm g₀ p y) univ := ContinuousSmoothFamilyOn.const (P:=P) (chartMetricForm g₀ p) (fun y _ => (chartMetricForm_contDiffOn g₀ p).contDiffAt (ht ▸ univ_mem)) intro n change Continuous (fun z : P × E => iteratedFDeriv ℝ n (fun y => chartMetricForm (q z.1) p y-chartMetricForm g₀ p y) z.2) simpa only [univ_prod_univ,continuousOn_univ,chartMetricForm,metricMatrixFormCLM_apply] using (hfm.sub hfb).jets n end YauCounterexamples end end OAI