| import OAI.Geometry.Arnold.ThreePoints |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ContDiff |
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| namespace ArnoldCounterexample |
| namespace ThreePoints |
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| open scoped Matrix |
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| open scoped Manifold RealInnerProductSpace |
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| abbrev E3 := EuclideanSpace ℝ (Fin 3) |
| abbrev S2 := Metric.sphere (0 : E3) 1 |
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| local instance : Fact (Module.finrank ℝ E3 = 2 + 1) := ⟨by simp [E3]⟩ |
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| def coords : E3 ≃L[ℝ] R3 := EuclideanSpace.equiv (Fin 3) ℝ |
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| def sphereEmbed (x : S2) : R3 := coords (x : E3) |
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| def spherePairEmbed : S2 × S2 → Pair := Prod.map sphereEmbed sphereEmbed |
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| def sphereF : S2 × S2 → ℝ := f ∘ spherePairEmbed |
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| lemma inner_coords (x y : E3) : inner ℝ x y = coords x ⬝ᵥ coords y := by |
| rw [EuclideanSpace.inner_eq_star_dotProduct] |
| simp only [coords, EuclideanSpace.equiv, star_trivial] |
| exact dotProduct_comm _ _ |
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| lemma norm_sq_coords (x : E3) : ‖x‖ ^ 2 = sqNorm (coords x) := by |
| rw [← real_inner_self_eq_norm_sq, inner_coords, dot_self] |
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| lemma sphereEmbed_mem (x : S2) : sqNorm (sphereEmbed x) = 1 := by |
| rw [sphereEmbed, ← norm_sq_coords] |
| have h : ‖(x : E3)‖ = 1 := norm_eq_of_mem_sphere x |
| rw [h]; norm_num |
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| lemma spherePairEmbed_mem (q : S2 × S2) : spherePairEmbed q ∈ sphereProduct := |
| ⟨sphereEmbed_mem q.1, sphereEmbed_mem q.2⟩ |
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| lemma sphereEmbed_injective : Function.Injective sphereEmbed := by |
| intro x y h |
| exact Subtype.ext (coords.injective h) |
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| lemma spherePairEmbed_injective : Function.Injective spherePairEmbed := |
| Function.Injective.prodMap sphereEmbed_injective sphereEmbed_injective |
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| lemma exists_sphereEmbed {v : R3} (hv : sqNorm v = 1) : |
| ∃ x : S2, sphereEmbed x = v := by |
| have hn : ‖coords.symm v‖ = 1 := by |
| have hs : ‖coords.symm v‖ ^ 2 = 1 := by |
| rw [norm_sq_coords, coords.apply_symm_apply, hv] |
| nlinarith [norm_nonneg (coords.symm v)] |
| refine ⟨⟨coords.symm v, ?_⟩, coords.apply_symm_apply v⟩ |
| simpa using hn |
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| lemma range_spherePairEmbed : Set.range spherePairEmbed = sphereProduct := by |
| ext q |
| constructor |
| · rintro ⟨x, rfl⟩ |
| exact spherePairEmbed_mem x |
| · rintro ⟨hx, hy⟩ |
| obtain ⟨x, hx⟩ := exists_sphereEmbed hx |
| obtain ⟨y, hy⟩ := exists_sphereEmbed hy |
| exact ⟨(x, y), Prod.ext hx hy⟩ |
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| lemma sphereEmbed_smooth : ContMDiff (𝓡 2) 𝓘(ℝ, R3) ∞ sphereEmbed := |
| coords.contDiff.contMDiff.comp contMDiff_coe_sphere |
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| lemma spherePairEmbed_smooth : |
| ContMDiff ((𝓡 2).prod (𝓡 2)) 𝓘(ℝ, Pair) ∞ spherePairEmbed := by |
| rw [modelWithCornersSelf_prod, ← chartedSpaceSelf_prod] |
| exact sphereEmbed_smooth.prodMap sphereEmbed_smooth |
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| theorem sphereF_smooth : ContMDiff ((𝓡 2).prod (𝓡 2)) 𝓘(ℝ, ℝ) ∞ sphereF := |
| f_contDiff.contMDiff.comp spherePairEmbed_smooth |
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| def sphereEmbedDeriv (x : S2) : TangentSpace (𝓡 2) x →L[ℝ] R3 := |
| coords.toContinuousLinearMap.comp |
| (mvfderiv (𝓡 2) (Subtype.val : S2 → E3) x) |
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| lemma sphereEmbed_mfderiv (x : S2) : |
| mfderiv (𝓡 2) 𝓘(ℝ, R3) sphereEmbed x = sphereEmbedDeriv x := by |
| change mfderiv (𝓡 2) 𝓘(ℝ, R3) |
| ((coords : E3 → R3) ∘ (Subtype.val : S2 → E3)) x = _ |
| rw [mfderiv_comp x (coords.contDiff.contMDiff.mdifferentiable (n := ∞) (by norm_num)).mdifferentiableAt |
| (((contMDiff_coe_sphere (m := ∞) (n := 2)).mdifferentiable (by norm_num))).mdifferentiableAt, |
| mfderiv_eq_fderiv, coords.fderiv] |
| rfl |
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| lemma sphereEmbedDeriv_range (x : S2) (v : R3) : |
| v ∈ (sphereEmbedDeriv x).range ↔ sphereEmbed x ⬝ᵥ v = 0 := by |
| constructor |
| · rintro ⟨w, rfl⟩ |
| have hw : mvfderiv (𝓡 2) (Subtype.val : S2 → E3) x w ∈ (ℝ ∙ (x : E3))ᗮ := by |
| rw [← range_mvfderiv_subtypeVal (n := 2) x] |
| exact ⟨w, rfl⟩ |
| have := Submodule.mem_orthogonal_singleton_iff_inner_right.mp hw |
| change coords (x : E3) ⬝ᵥ coords (mvfderiv (𝓡 2) (Subtype.val : S2 → E3) x w) = 0 |
| rwa [← inner_coords] |
| · intro hv |
| have hw : coords.symm v ∈ (ℝ ∙ (x : E3))ᗮ := by |
| rw [Submodule.mem_orthogonal_singleton_iff_inner_right, inner_coords, |
| coords.apply_symm_apply] |
| exact hv |
| rw [← range_mvfderiv_subtypeVal (n := 2) x] at hw |
| obtain ⟨w, hw⟩ := hw |
| refine ⟨w, ?_⟩ |
| change coords (mvfderiv (𝓡 2) (Subtype.val : S2 → E3) x w) = v |
| exact (congrArg coords hw).trans (coords.apply_symm_apply v) |
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| lemma spherePairEmbed_mfderiv (q : S2 × S2) : |
| mfderiv ((𝓡 2).prod (𝓡 2)) 𝓘(ℝ, Pair) spherePairEmbed q = |
| (sphereEmbedDeriv q.1).prodMap (sphereEmbedDeriv q.2) := by |
| rw [modelWithCornersSelf_prod, ← chartedSpaceSelf_prod] |
| change mfderiv ((𝓡 2).prod (𝓡 2)) (𝓘(ℝ, R3).prod 𝓘(ℝ, R3)) |
| (Prod.map sphereEmbed sphereEmbed) q = _ |
| rw [mfderiv_prodMap |
| (sphereEmbed_smooth.mdifferentiable (by norm_num)).mdifferentiableAt |
| (sphereEmbed_smooth.mdifferentiable (by norm_num)).mdifferentiableAt, |
| sphereEmbed_mfderiv, sphereEmbed_mfderiv] |
| rfl |
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| lemma sphereF_mfderiv (q : S2 × S2) : |
| mfderiv ((𝓡 2).prod (𝓡 2)) 𝓘(ℝ, ℝ) sphereF q = |
| (df (spherePairEmbed q)).comp |
| ((sphereEmbedDeriv q.1).prodMap (sphereEmbedDeriv q.2)) := by |
| unfold sphereF |
| rw [mfderiv_comp q |
| (f_contDiff.contMDiff.mdifferentiable (by norm_num)).mdifferentiableAt |
| (spherePairEmbed_smooth.mdifferentiable (by norm_num)).mdifferentiableAt, |
| mfderiv_eq_fderiv, (f_hasFDerivAt _).fderiv] |
| rw [spherePairEmbed_mfderiv] |
| rfl |
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| lemma sphereF_critical_iff (q : S2 × S2) : |
| mfderiv ((𝓡 2).prod (𝓡 2)) 𝓘(ℝ, ℝ) sphereF q = 0 ↔ |
| TangentCritical (spherePairEmbed q) := by |
| rw [sphereF_mfderiv] |
| constructor |
| · intro h |
| refine ⟨spherePairEmbed_mem q, ?_⟩ |
| intro u v hu hv |
| obtain ⟨a, ha⟩ := (sphereEmbedDeriv_range q.1 u).mpr hu |
| obtain ⟨b, hb⟩ := (sphereEmbedDeriv_range q.2 v).mpr hv |
| have hh := DFunLike.congr_fun h (a, b) |
| change df (spherePairEmbed q) (sphereEmbedDeriv q.1 a, sphereEmbedDeriv q.2 b) = 0 at hh |
| change sphereEmbedDeriv q.1 a = u at ha |
| change sphereEmbedDeriv q.2 b = v at hb |
| rw [ha, hb] at hh |
| simpa only [(f_hasFDerivAt _).fderiv] using hh |
| · rintro ⟨_, h⟩ |
| apply ContinuousLinearMap.ext |
| intro w |
| change df (spherePairEmbed q) (sphereEmbedDeriv q.1 w.1, sphereEmbedDeriv q.2 w.2) = 0 |
| rw [← (f_hasFDerivAt _).fderiv] |
| exact h _ _ ((sphereEmbedDeriv_range q.1 _).mp ⟨w.1, rfl⟩) |
| ((sphereEmbedDeriv_range q.2 _).mp ⟨w.2, rfl⟩) |
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| / |
| theorem sphereF_critical_encard : |
| {q : S2 × S2 | mfderiv ((𝓡 2).prod (𝓡 2)) 𝓘(ℝ, ℝ) sphereF q = 0}.encard = 3 := by |
| have he : {q : S2 × S2 | mfderiv ((𝓡 2).prod (𝓡 2)) 𝓘(ℝ, ℝ) sphereF q = 0} = |
| spherePairEmbed ⁻¹' {q : Pair | TangentCritical q} := by |
| ext q |
| exact sphereF_critical_iff q |
| rw [he, Set.encard_preimage_of_injective_subset_range spherePairEmbed_injective] |
| · exact tangentCritical_encard |
| · rw [range_spherePairEmbed] |
| exact fun _ h ↦ h.1 |
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| end ThreePoints |
| end ArnoldCounterexample |
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| end |
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| end OAI |
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