File size: 2,923 Bytes
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 | import OAI.Geometry.EinsteinFour.CP2NonorthogonalFixed
import OAI.Geometry.EinsteinFour.CP2OddOrthogonality
namespace OAI
/-! Every involution of CP2 preserving its Fubini--Study distance has a
fixed point. The proof uses only the three coordinate lines and the unique
central line of a nonorthogonal orbit. -/
namespace PositiveEinsteinFour
open EinsteinFour
noncomputable section
private theorem cp2_not_everywhere_orthogonal (f : ComplexProjective2 → ComplexProjective2)
(hinv : Function.Involutive f)
(hpres : ∀ p q, cp2Overlap (f p) (f q) = cp2Overlap p q)
(hzero : ∀ p, cp2Overlap p (f p) = 0) : False := by
let e (i : Fin 3) : CP2Vector := EuclideanSpace.single i 1
have hen (i : Fin 3) : ‖e i‖ = 1 := by simp [e]
have he0 (i : Fin 3) : e i ≠ 0 := by
intro h
have hn := hen i
rw [h, norm_zero] at hn
exact zero_ne_one hn
let p (i : Fin 3) : ComplexProjective2 := Projectivization.mk ℂ (e i) (he0 i)
let v (i : Fin 3) := cp2UnitRep (f (p i))
have hvn (i : Fin 3) : ‖v i‖ = 1 := cp2UnitRep_norm _
have hcoord (i j : Fin 3) : cp2Overlap (p i) (f (p j)) = ‖v j i‖ := by
rw [cp2Overlap_of_unit_representatives (p i) (f (p j)) (e i) (v j)
(he0 i) (cp2UnitRep_ne_zero _) (hen i) (hvn j) rfl (cp2UnitRep_mk _)]
simp [e, EuclideanSpace.inner_single_left]
have hdiag (i : Fin 3) : v i i = 0 := by
apply norm_eq_zero.mp
rw [← hcoord, hzero]
have horth (i j : Fin 3) (hij : i ≠ j) : inner ℂ (v i) (v j) = 0 := by
apply norm_eq_zero.mp
change cp2Overlap (f (p i)) (f (p j)) = 0
rw [hpres, cp2Overlap_of_unit_representatives (p i) (p j) (e i) (e j)
(he0 i) (he0 j) (hen i) (hen j) rfl rfl]
simp [e, EuclideanSpace.inner_single_left, hij]
have hsym (i j : Fin 3) : ‖v i j‖ = ‖v j i‖ := by
rw [← hcoord, ← hcoord]
calc
cp2Overlap (p j) (f (p i)) = cp2Overlap (f (p j)) (p i) := by
simpa only [hinv (p i)] using (hpres (p j) (f (p i))).symm
_ = cp2Overlap (p i) (f (p j)) := cp2Overlap_comm _ _
exact cp2_odd_orthogonality_obstruction v hvn horth hdiag hsym
theorem cp2_involutive_overlap_fixed (f : ComplexProjective2 → ComplexProjective2)
(hinv : Function.Involutive f)
(hpres : ∀ p q, cp2Overlap (f p) (f q) = cp2Overlap p q) :
∃ p, f p = p := by
by_contra hn
apply cp2_not_everywhere_orthogonal f hinv hpres
intro p
apply le_antisymm _ (cp2Overlap_nonneg _ _)
by_contra hp
exact hn (cp2_nonorthogonal_fixed f hinv hpres p (lt_of_not_ge hp))
theorem cp2_involutive_isometry_fixed (f : ComplexProjective2 → ComplexProjective2)
(hinv : Function.Involutive f)
(hdist : ∀ p q, fubiniStudyDistance (f p) (f q) = fubiniStudyDistance p q) :
∃ p, f p = p := by
apply cp2_involutive_overlap_fixed f hinv
intro p q
exact (cp2Overlap_eq_iff_distance_eq _ _ _ _).mpr (hdist p q)
end
end PositiveEinsteinFour
end OAI
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