| import OAI.Geometry.EinsteinFour.CP2NonorthogonalFixed |
| import OAI.Geometry.EinsteinFour.CP2OddOrthogonality |
|
|
| namespace OAI |
|
|
| /-! Every involution of CP2 preserving its Fubini |
| 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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|
