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785c727 | 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 | import OAI.Geometry.EinsteinFour.CP2PairUniqueness
namespace OAI
/-! The unit representatives needed for the projective fixed-point argument. -/
namespace PositiveEinsteinFour
open EinsteinFour
noncomputable section
theorem cp2Overlap_of_unit_representatives (p q : ComplexProjective2)
(x y : CP2Vector) (hx : x ≠ 0) (hy : y ≠ 0)
(hnx : ‖x‖ = 1) (hny : ‖y‖ = 1)
(hpx : Projectivization.mk ℂ x hx = p) (hqy : Projectivization.mk ℂ y hy = q) :
cp2Overlap p q = ‖inner ℂ x y‖ := by
rw [← hpx, ← hqy, cp2Overlap_mk, hnx, hny, one_mul, div_one]
theorem cp2Overlap_eq_one_iff (p q : ComplexProjective2) :
cp2Overlap p q = 1 ↔ p = q := by
constructor
· intro h
have hCS : ‖inner ℂ (cp2UnitRep p) (cp2UnitRep q)‖ =
‖cp2UnitRep p‖ * ‖cp2UnitRep q‖ := by
simpa only [cp2Overlap, cp2UnitRep_norm, one_mul] using h
obtain ⟨a, _, ha⟩ :=
(norm_inner_eq_norm_iff (cp2UnitRep_ne_zero p) (cp2UnitRep_ne_zero q)).mp hCS
rw [← cp2UnitRep_mk p, ← cp2UnitRep_mk q]
apply (Projectivization.mk_eq_mk_iff' ℂ _ _ _ _).mpr
exact ⟨a⁻¹, by rw [ha, smul_smul]; simp_all⟩
· rintro rfl
exact cp2Overlap_self p
theorem cp2_unit_aligned (p q : ComplexProjective2) (h : 0 < cp2Overlap p q) :
∃ v : CP2Vector, ∃ hv : v ≠ 0,
‖v‖ = 1 ∧ Projectivization.mk ℂ v hv = q ∧
inner ℂ (cp2UnitRep p) v = (cp2Overlap p q : ℂ) := by
let z := inner ℂ (cp2UnitRep p) (cp2UnitRep q)
have hz : z ≠ 0 := by
intro hz
change 0 < ‖z‖ at h
rw [hz, norm_zero] at h
exact (lt_irrefl 0) h
let a : ℂ := (‖z‖ : ℂ) / z
have ha : a ≠ 0 := div_ne_zero (by exact_mod_cast (norm_ne_zero_iff.mpr hz)) hz
have han : ‖a‖ = 1 := by
simp [a, Complex.norm_real, norm_ne_zero_iff.mpr hz]
let v : CP2Vector := a • cp2UnitRep q
have hv : v ≠ 0 := smul_ne_zero ha (cp2UnitRep_ne_zero q)
refine ⟨v, hv, ?_, ?_, ?_⟩
· simp only [v, norm_smul, han, cp2UnitRep_norm, mul_one]
· rw [← cp2UnitRep_mk q]
exact (Projectivization.mk_eq_mk_iff' ℂ _ _ _ _).mpr ⟨a, rfl⟩
· change inner ℂ (cp2UnitRep p) (a • cp2UnitRep q) = (‖z‖ : ℂ)
rw [inner_smul_right]
change ((‖z‖ : ℂ) / z) * z = _
exact div_mul_cancel₀ _ hz
theorem cp2_aligned_center (p q : CP2Vector) (c : ℝ)
(hp : ‖p‖ = 1) (hq : ‖q‖ = 1) (hc : inner ℂ p q = (c : ℂ))
(hc0 : 0 < c) :
∃ u : CP2Vector, ‖u‖ = 1 ∧
‖inner ℂ p u‖ ^ 2 + ‖inner ℂ q u‖ ^ 2 = 1 + c := by
have hqp : inner ℂ q p = (c : ℂ) := by rw [← inner_conj_symm, hc]; simp
have hs : ‖p + q‖ ^ 2 = 2 + 2 * c := by
rw [norm_add_sq (𝕜 := ℂ), hp, hq, hc]
simp
ring
have hn : ‖p + q‖ ≠ 0 := by intro h; rw [h] at hs; nlinarith
let u : CP2Vector := ((‖p + q‖⁻¹ : ℝ) : ℂ) • (p + q)
have hpi : inner ℂ p (p + q) = ((1 + c : ℝ) : ℂ) := by
simp [inner_self_eq_norm_sq_to_K, hp, hc]
have hqi : inner ℂ q (p + q) = ((1 + c : ℝ) : ℂ) := by
simp [inner_self_eq_norm_sq_to_K, hq, hqp, add_comm]
have hpn : ‖inner ℂ p u‖ = ‖p + q‖⁻¹ * (1 + c) := by
simp only [u, inner_smul_right, hpi, norm_mul, Complex.norm_real]
rw [Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _)),
Real.norm_of_nonneg (by linarith)]
have hqn : ‖inner ℂ q u‖ = ‖p + q‖⁻¹ * (1 + c) := by
simp only [u, inner_smul_right, hqi, norm_mul, Complex.norm_real]
rw [Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _)),
Real.norm_of_nonneg (by linarith)]
refine ⟨u, ?_, ?_⟩
· simp only [u, norm_smul, Complex.norm_real,
Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _)), inv_mul_cancel₀ hn]
· rw [hpn, hqn]
field_simp
nlinarith
end
end PositiveEinsteinFour
end OAI
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