| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open scoped Manifold ContDiff BigOperators ENNReal |
| open Bundle |
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| namespace EinsteinFour |
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| abbrev FourSpace := EuclideanSpace ℝ (Fin 4) |
| abbrev I4 := 𝓘(ℝ, FourSpace) |
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| def coordinateVector (i : Fin 4) : FourSpace := EuclideanSpace.single i 1 |
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| abbrev RoundSphere4 := {x : EuclideanSpace ℝ (Fin 5) |
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| abbrev ComplexProjective2 := Projectivization ℂ (EuclideanSpace ℂ (Fin 3)) |
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| abbrev RealProjective4 := Projectivization ℝ (EuclideanSpace ℝ (Fin 5)) |
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| def roundDistance (p q : RoundSphere4) : ℝ := |
| Real.arccos (inner ℝ p.val q.val) |
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| def fubiniStudyDistance (p q : ComplexProjective2) : ℝ := |
| Real.arccos (‖inner ℂ p.rep q.rep‖ / (‖p.rep‖ * ‖q.rep‖)) |
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| def realProjectiveDistance (p q : RealProjective4) : ℝ := |
| Real.arccos (‖inner ℝ p.rep q.rep‖ / (‖p.rep‖ * ‖q.rep‖)) |
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| section Metric |
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| variable (M : Type*) [TopologicalSpace M] [ChartedSpace FourSpace M] |
| [IsManifold I4 ∞ M] |
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| |
| abbrev SmoothMetric := |
| ContMDiffRiemannianMetric I4 ∞ FourSpace (TangentSpace I4 : M → Type _) |
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| variable {M} |
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| def metricInChart (g : SmoothMetric M) (c : M) (z : FourSpace) : |
| Matrix (Fin 4) (Fin 4) ℝ := |
| let e := chartAt FourSpace c |
| let d := mfderiv I4 I4 e.symm z |
| fun i j => g.inner (e.symm z) (d (coordinateVector i)) (d (coordinateVector j)) |
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| |
| def partialDeriv (f : FourSpace → ℝ) (i : Fin 4) (z : FourSpace) : ℝ := |
| fderiv ℝ f z (coordinateVector i) |
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| def christoffel (g : SmoothMetric M) (c : M) (z : FourSpace) |
| (k i j : Fin 4) : ℝ := |
| (1 / 2 : ℝ) * ∑ l : Fin 4, (metricInChart g c z)⁻¹ k l * |
| (partialDeriv (fun w => metricInChart g c w j l) i z + |
| partialDeriv (fun w => metricInChart g c w i l) j z - |
| partialDeriv (fun w => metricInChart g c w i j) l z) |
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| def curvatureComponent (g : SmoothMetric M) (c : M) (z : FourSpace) |
| (l i j k : Fin 4) : ℝ := |
| partialDeriv (fun w => christoffel g c w l j k) i z - |
| partialDeriv (fun w => christoffel g c w l i k) j z + |
| ∑ p : Fin 4, |
| (christoffel g c z l i p * christoffel g c z p j k - |
| christoffel g c z l j p * christoffel g c z p i k) |
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| |
| def ricciInChart (g : SmoothMetric M) (c : M) (z : FourSpace) |
| (i j : Fin 4) : ℝ := |
| ∑ k : Fin 4, curvatureComponent g c z k k i j |
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| def sectionalNumerator (g : SmoothMetric M) (c : M) (z : FourSpace) |
| (u v : FourSpace) : ℝ := |
| ∑ l : Fin 4, ∑ m : Fin 4, ∑ i : Fin 4, ∑ j : Fin 4, ∑ k : Fin 4, |
| metricInChart g c z l m * curvatureComponent g c z l i j k * |
| u i * v j * v k * u m |
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| def IsEinstein (g : SmoothMetric M) : Prop := |
| ∃ lam : ℝ, ∀ (c : M) (i j : Fin 4), |
| ricciInChart g c (chartAt FourSpace c c) i j = |
| lam * metricInChart g c (chartAt FourSpace c c) i j |
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| def HasPositiveSectionalCurvature (g : SmoothMetric M) : Prop := |
| ∀ (c : M) (u v : FourSpace), LinearIndependent ℝ ![u, v] → |
| 0 < sectionalNumerator g c (chartAt FourSpace c c) u v |
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| def scaledEDistance (g : SmoothMetric M) (a : ℝ) (x y : M) : ℝ≥0∞ := |
| letI : RiemannianBundle (TangentSpace I4 : M → Type _) := ⟨g.toRiemannianMetric⟩ |
| ENNReal.ofReal (Real.sqrt a) * Manifold.riemannianEDist I4 x y |
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| def ScaledIsometricTo {N : Type*} (g : SmoothMetric M) (a : ℝ) |
| (modelDistance : N → N → ℝ) : Prop := |
| ∃ f : M ≃ N, ∀ x y : M, |
| ENNReal.ofReal (modelDistance (f x) (f y)) = scaledEDistance g a x y |
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| end Metric |
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| |
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| section SphereInvolution |
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| variable {E : Type*} [NormedAddCommGroup E] |
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| abbrev UnitSphere (E : Type*) [NormedAddCommGroup E] := {x : E |
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| def antipode (x : UnitSphere E) : UnitSphere E := |
| ⟨-x.val, by simpa using x.property⟩ |
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| @[simp] theorem antipode_val (x : UnitSphere E) : (antipode x).val = -x.val := rfl |
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| @[simp] theorem antipode_antipode (x : UnitSphere E) : antipode (antipode x) = x := by |
| apply Subtype.ext |
| simp |
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| theorem unitSphere_ne_zero (x : UnitSphere E) : x.val ≠ 0 := by |
| intro h |
| have hx := x.property |
| simp [h] at hx |
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| variable [InnerProductSpace ℝ E] |
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| theorem antipode_ne_self (x : UnitSphere E) : antipode x ≠ x := by |
| intro h |
| have h' : -x.val = x.val := congrArg Subtype.val h |
| have hz : x.val = 0 := by |
| have hh : (2 : ℝ) • x.val = 0 := by |
| calc |
| (2 : ℝ) • x.val = x.val + x.val := by module |
| _ = -x.val + x.val := by rw [h'] |
| _ = 0 := neg_add_cancel x.val |
| exact (smul_eq_zero.mp hh).resolve_left (by norm_num) |
| exact unitSphere_ne_zero x hz |
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| theorem sphere_involution_antipodal |
| (τ : UnitSphere E → UnitSphere E) |
| (hτ : Function.Involutive τ) |
| (hinner : ∀ x y, inner ℝ (τ x).val (τ y).val = inner ℝ x.val y.val) |
| (hfree : ∀ x, τ x ≠ x) : |
| ∀ x, τ x = antipode x := by |
| intro x |
| apply Subtype.ext |
| change (τ x).val = -x.val |
| by_contra hneg |
| have hv : x.val + (τ x).val ≠ 0 := by |
| intro hz |
| exact hneg (eq_neg_of_add_eq_zero_right hz) |
| let r : ℝ := ‖x.val + (τ x).val‖ |
| have hr : 0 < r := norm_pos_iff.mpr hv |
| let z : UnitSphere E := |
| ⟨r⁻¹ • (x.val + (τ x).val), by |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hr)] |
| exact inv_mul_cancel₀ (ne_of_gt hr)⟩ |
| have hzx := hinner z x |
| have hzτx := hinner z (τ x) |
| rw [hτ x] at hzτx |
| have hzinner : inner ℝ (τ z).val z.val = inner ℝ z.val z.val := by |
| change inner ℝ (τ z).val (r⁻¹ • (x.val + (τ x).val)) = |
| inner ℝ z.val (r⁻¹ • (x.val + (τ x).val)) |
| simp only [inner_smul_right, inner_add_right] |
| rw [hzx, hzτx, add_comm] |
| have hzfixed : τ z = z := by |
| apply Subtype.ext |
| apply sub_eq_zero.mp |
| apply (inner_self_eq_zero (𝕜 := ℝ)).mp |
| have hzinner' : inner ℝ z.val (τ z).val = inner ℝ z.val z.val := |
| (real_inner_comm (τ z).val z.val).trans hzinner |
| simp only [inner_sub_left, inner_sub_right, hinner z z, hzinner, hzinner', |
| sub_self] |
| exact hfree z hzfixed |
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| end SphereInvolution |
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| theorem round_distance_preserving_inner |
| (τ : RoundSphere4 → RoundSphere4) |
| (h : ∀ x y, roundDistance (τ x) (τ y) = roundDistance x y) : |
| ∀ x y, inner ℝ (τ x).val (τ y).val = inner ℝ x.val y.val := by |
| have bounds (x y : RoundSphere4) : |
| -(1 : ℝ) ≤ inner ℝ x.val y.val ∧ inner ℝ x.val y.val ≤ 1 := by |
| apply abs_le.mp |
| simpa [x.property, y.property] using abs_real_inner_le_norm x.val y.val |
| intro x y |
| have heq := congrArg Real.cos (h x y) |
| dsimp only [roundDistance] at heq |
| rw [Real.cos_arccos (bounds (τ x) (τ y)).1 (bounds (τ x) (τ y)).2, |
| Real.cos_arccos (bounds x y).1 (bounds x y).2] at heq |
| exact heq |
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| theorem free_round_involution_eq_antipode |
| (τ : RoundSphere4 → RoundSphere4) |
| (hτ : Function.Involutive τ) |
| (hDistance : ∀ x y, roundDistance (τ x) (τ y) = roundDistance x y) |
| (hfree : ∀ x, τ x ≠ x) : τ = antipode := by |
| funext x |
| exact sphere_involution_antipodal τ hτ |
| (round_distance_preserving_inner τ hDistance) hfree x |
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| section ProjectiveDistance |
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| variable {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] |
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| theorem normalized_inner_smul (x y : E) (a b : 𝕜) (ha : a ≠ 0) (hb : b ≠ 0) : |
| ‖inner 𝕜 (a • x) (b • y)‖ / (‖a • x‖ * ‖b • y‖) = |
| ‖inner 𝕜 x y‖ / (‖x‖ * ‖y‖) := by |
| simp only [inner_smul_left, inner_smul_right, norm_mul, RCLike.norm_conj, norm_smul] |
| calc |
| ‖b‖ * (‖a‖ * ‖inner 𝕜 x y‖) / (‖a‖ * ‖x‖ * (‖b‖ * ‖y‖)) = |
| (‖a‖ * ‖b‖) * ‖inner 𝕜 x y‖ / ((‖a‖ * ‖b‖) * (‖x‖ * ‖y‖)) := by |
| congr 1 <;> ring |
| _ = _ := mul_div_mul_left _ _ (mul_ne_zero (norm_ne_zero_iff.mpr ha) |
| (norm_ne_zero_iff.mpr hb)) |
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| theorem normalized_inner_mk_rep (x y : E) (hx : x ≠ 0) (hy : y ≠ 0) : |
| ‖inner 𝕜 (Projectivization.mk 𝕜 x hx).rep (Projectivization.mk 𝕜 y hy).rep‖ / |
| (‖(Projectivization.mk 𝕜 x hx).rep‖ * ‖(Projectivization.mk 𝕜 y hy).rep‖) = |
| ‖inner 𝕜 x y‖ / (‖x‖ * ‖y‖) := by |
| obtain ⟨a, ha⟩ := Projectivization.exists_smul_eq_mk_rep 𝕜 x hx |
| obtain ⟨b, hb⟩ := Projectivization.exists_smul_eq_mk_rep 𝕜 y hy |
| rw [← ha, ← hb] |
| exact normalized_inner_smul x y (a : 𝕜) (b : 𝕜) a.ne_zero b.ne_zero |
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| end ProjectiveDistance |
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| theorem fubiniStudyDistance_mk (x y : EuclideanSpace ℂ (Fin 3)) |
| (hx : x ≠ 0) (hy : y ≠ 0) : |
| fubiniStudyDistance (Projectivization.mk ℂ x hx) (Projectivization.mk ℂ y hy) = |
| Real.arccos (‖inner ℂ x y‖ / (‖x‖ * ‖y‖)) := by |
| rw [fubiniStudyDistance, normalized_inner_mk_rep] |
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| theorem realProjectiveDistance_mk (x y : EuclideanSpace ℝ (Fin 5)) |
| (hx : x ≠ 0) (hy : y ≠ 0) : |
| realProjectiveDistance (Projectivization.mk ℝ x hx) (Projectivization.mk ℝ y hy) = |
| Real.arccos (‖inner ℝ x y‖ / (‖x‖ * ‖y‖)) := by |
| rw [realProjectiveDistance, normalized_inner_mk_rep] |
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| section AntipodalQuotient |
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| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
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| def sphereToProjective (x : UnitSphere E) : Projectivization ℝ E := |
| Projectivization.mk ℝ x.val (unitSphere_ne_zero x) |
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| theorem sphereToProjective_surjective : |
| Function.Surjective (sphereToProjective : UnitSphere E → Projectivization ℝ E) := by |
| intro p |
| have hp : 0 < ‖p.rep‖ := norm_pos_iff.mpr p.rep_nonzero |
| let x : UnitSphere E := ⟨‖p.rep‖⁻¹ • p.rep, by |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hp)] |
| exact inv_mul_cancel₀ (ne_of_gt hp)⟩ |
| refine ⟨x, ?_⟩ |
| change Projectivization.mk ℝ x.val (unitSphere_ne_zero x) = p |
| conv_rhs => rw [← p.mk_rep] |
| refine (Projectivization.mk_eq_mk_iff' ℝ _ _ _ _).mpr ⟨‖p.rep‖⁻¹, rfl⟩ |
| |
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| theorem sphereToProjective_eq_iff (x y : UnitSphere E) : |
| sphereToProjective x = sphereToProjective y ↔ x = y ∨ x = antipode y := by |
| rw [sphereToProjective, sphereToProjective, Projectivization.mk_eq_mk_iff'] |
| constructor |
| · rintro ⟨a, ha⟩ |
| have habs := congrArg norm ha |
| simp only [norm_smul, Real.norm_eq_abs, x.property, y.property, mul_one] at habs |
| have ha' : a = 1 ∨ a = -1 := by |
| rcases le_total 0 a with h | h |
| · left |
| rwa [abs_of_nonneg h] at habs |
| · right |
| rw [abs_of_nonpos h] at habs |
| linarith |
| rcases ha' with ha' | ha' |
| · left |
| apply Subtype.ext |
| simpa [ha'] using ha.symm |
| · right |
| apply Subtype.ext |
| simpa [ha'] using ha.symm |
| · rintro (h | h) |
| · exact ⟨1, by simpa using (congrArg Subtype.val h).symm⟩ |
| · exact ⟨-1, by simpa using (congrArg Subtype.val h).symm⟩ |
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| def antipodalSetoid (E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] : |
| Setoid (UnitSphere E) where |
| r x y := x = y ∨ x = antipode y |
| iseqv := { |
| refl := fun _ => Or.inl rfl |
| symm := fun h => sphereToProjective_eq_iff _ _ |>.mp |
| ((sphereToProjective_eq_iff _ _).mpr h).symm |
| trans := fun h₁ h₂ => (sphereToProjective_eq_iff _ _).mp |
| (((sphereToProjective_eq_iff _ _).mpr h₁).trans |
| ((sphereToProjective_eq_iff _ _).mpr h₂)) } |
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| def antipodalQuotientEquiv : Quotient (antipodalSetoid E) ≃ Projectivization ℝ E := |
| Equiv.ofBijective |
| (Quotient.lift sphereToProjective fun x y h => (sphereToProjective_eq_iff x y).mpr h) |
| (by |
| constructor |
| · intro x y h |
| induction x using Quotient.inductionOn with | h x => |
| induction y using Quotient.inductionOn with | h y => |
| exact Quotient.sound ((sphereToProjective_eq_iff x y).mp h) |
| · intro p |
| obtain ⟨x, hx⟩ := sphereToProjective_surjective (E := E) p |
| exact ⟨Quotient.mk _ x, hx⟩) |
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| @[simp] theorem antipodalQuotientEquiv_mk (x : UnitSphere E) : |
| antipodalQuotientEquiv (Quotient.mk (antipodalSetoid E) x) = sphereToProjective x := rfl |
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| end AntipodalQuotient |
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| |
| theorem realProjectiveDistance_sphere (x y : RoundSphere4) : |
| realProjectiveDistance (sphereToProjective x) (sphereToProjective y) = |
| Real.arccos |inner ℝ x.val y.val| := by |
| rw [sphereToProjective, sphereToProjective, realProjectiveDistance_mk, |
| x.property, y.property, mul_one, div_one, Real.norm_eq_abs] |
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| theorem realProjectiveDistance_eq_min (x y : RoundSphere4) : |
| realProjectiveDistance (sphereToProjective x) (sphereToProjective y) = |
| min (roundDistance x y) (roundDistance x (antipode y)) := by |
| rw [realProjectiveDistance_sphere] |
| simp only [roundDistance, antipode_val, inner_neg_right] |
| by_cases h : 0 ≤ inner ℝ x.val y.val |
| · rw [abs_of_nonneg h, min_eq_left] |
| exact Real.antitone_arccos (by linarith) |
| · rw [abs_of_neg (lt_of_not_ge h), min_eq_right] |
| exact Real.antitone_arccos (by linarith) |
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| theorem free_round_involution_projective_fibers_and_distance |
| (τ : RoundSphere4 → RoundSphere4) |
| (hτ : Function.Involutive τ) |
| (hDistance : ∀ x y, roundDistance (τ x) (τ y) = roundDistance x y) |
| (hfree : ∀ x, τ x ≠ x) : |
| (∀ x y, sphereToProjective x = sphereToProjective y ↔ x = y ∨ x = τ y) ∧ |
| (∀ x y, realProjectiveDistance (sphereToProjective x) (sphereToProjective y) = |
| min (roundDistance x y) (roundDistance x (τ y))) := by |
| rw [free_round_involution_eq_antipode τ hτ hDistance hfree] |
| exact ⟨sphereToProjective_eq_iff, realProjectiveDistance_eq_min⟩ |
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| end EinsteinFour |
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| end |
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| end OAI |
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