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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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Geometry.EinsteinFour.AngularRotation
namespace OAI
/-! The pointwise and averaged normal Jacobi curvature form. -/
namespace PositiveEinsteinFour.Angular
open Matrix MeasureTheory
noncomputable section
abbrev Mat3 := Geometry.Mat3
def squareNorm (T : Mat3) : ℝ := ∑ i, ∑ j, T i j ^ 2
def pairing (T U : Mat3) : ℝ := ∑ i, ∑ j, T i j * U i j
def rotated (U : Mat3) (q : Directions) : Mat3 :=
mixingMatrix q * U * (mixingMatrix q)ᵀ
/-- The curvature deviation in the normal basis `I_s q`. -/
def normalDeviation (T U : Mat3) (q : Directions) : Mat3 :=
(1 / 2 : ℝ) • (T + rotated U q)
theorem squareNorm_eq_trace (T : Mat3) : squareNorm T = (Tᵀ * T).trace := by
simp only [squareNorm, Matrix.trace, Matrix.diag, Matrix.mul_apply,
Matrix.transpose_apply]
rw [Finset.sum_comm]
simp only [pow_two]
theorem squareNorm_conjugate (O U : Mat3) (hO : O * Oᵀ = 1) :
squareNorm (O * U * Oᵀ) = squareNorm U := by
have hOT : Oᵀ * O = 1 := mul_eq_one_comm.mp hO
rw [squareNorm_eq_trace, squareNorm_eq_trace]
have hp : (O * U * Oᵀ)ᵀ * (O * U * Oᵀ) = O * (Uᵀ * U) * Oᵀ := by
simp only [Matrix.transpose_mul, Matrix.transpose_transpose]
calc
(O * (Uᵀ * Oᵀ)) * (O * U * Oᵀ) = O * Uᵀ * (Oᵀ * O) * U * Oᵀ := by
noncomm_ring
_ = _ := by rw [hOT, mul_one]; noncomm_ring
rw [hp, Matrix.trace_mul_cycle, hOT, one_mul]
theorem squareNorm_rotated (U : Mat3) (q : Directions) :
squareNorm (rotated U q) = squareNorm U :=
squareNorm_conjugate _ _ (mixing_orthogonal q)
private theorem norm_add_half (T U : Fin 3 → Fin 3 → ℝ) :
(∑ i, ∑ j, ((1 / 2 : ℝ) * (T i j + U i j)) ^ 2) =
((∑ i, ∑ j, T i j ^ 2) + (∑ i, ∑ j, U i j ^ 2) +
2 * (∑ i, ∑ j, T i j * U i j)) / 4 := by
have hp (i j : Fin 3) : ((1 / 2 : ℝ) * (T i j + U i j)) ^ 2 =
(T i j ^ 2 + U i j ^ 2 + 2 * (T i j * U i j)) / 4 := by ring
simp_rw [hp, ← Finset.sum_div, Finset.sum_add_distrib, ← Finset.mul_sum]
theorem squareNorm_normalDeviation (T U : Mat3) (q : Directions) :
squareNorm (normalDeviation T U q) =
(squareNorm T + squareNorm U + 2 * pairing T (rotated U q)) / 4 := by
rw [← squareNorm_rotated U q]
simpa only [squareNorm, pairing, normalDeviation, Matrix.smul_apply,
Matrix.add_apply, smul_eq_mul] using! norm_add_half T (rotated U q)
private theorem pairing_delta_sum (T U : Fin 3 → Fin 3 → ℝ) :
(∑ i, ∑ j, ∑ k, ∑ l, T i j * U k l *
(if i = j then (if k = l then (1 / 3 : ℝ) else 0) else 0)) =
((∑ i, T i i) * (∑ k, U k k)) / 3 := by
simp [mul_ite, div_eq_mul_inv, Finset.sum_mul, Finset.mul_sum, mul_assoc]
rw [Finset.sum_comm]
theorem integral_mixed_pairing (h : FollandMoments) (T U : Mat3)
(hT : T.trace = 0) (hU : U.trace = 0) :
(∫ q, pairing T (rotated U q) ∂sphereMeasure) = 0 := by
have he : (fun q => pairing T (rotated U q)) = fun q =>
∑ i, ∑ j, ∑ k, ∑ l, T i j * U k l * (mixing i k q * mixing j l q) := by
funext q
simp only [pairing, rotated, Matrix.mul_apply, Matrix.transpose_apply,
Finset.sum_mul, Finset.mul_sum]
simp only [mixingMatrix]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro k _
apply Finset.sum_congr rfl
intro l _
ring
rw [he, integral_sum4 h _ (fun _ _ _ _ => by unfold mixing quadratic; fun_prop)]
simp_rw [integral_const_mul, integral_mixing_mul h]
have ht : ∑ i, T i i = 0 := hT
have hu : ∑ i, U i i = 0 := hU
have hs := pairing_delta_sum T U
rw [ht, hu] at hs
simpa only [zero_mul, zero_div] using! hs
/-- The first identity in the manuscript's geodesic averages, on each
individual unit tangent sphere. -/
theorem integral_squareNorm_normalDeviation (h : FollandMoments) (T U : Mat3)
(hT : T.trace = 0) (hU : U.trace = 0) :
(∫ q, squareNorm (normalDeviation T U q) ∂sphereMeasure) =
(squareNorm T + squareNorm U) / 4 := by
let : IsProbabilityMeasure sphereMeasure := h.probability
have hc : Continuous (fun q => pairing T (rotated U q)) := by
simp only [pairing, rotated, Matrix.mul_apply, Matrix.transpose_apply]
simp only [mixingMatrix, mixing, quadratic]
fun_prop
have hi := continuous_integrable h hc
have he : (fun q => squareNorm (normalDeviation T U q)) =
fun q => (squareNorm T + squareNorm U + 2 * pairing T (rotated U q)) / 4 :=
funext (squareNorm_normalDeviation T U)
rw [he, integral_div, integral_add (integrable_const _) (hi.const_mul 2),
integral_const_mul, integral_mixed_pairing h T U hT hU]
simp
theorem integral_squareNorm_normalDeviation_normalized (h : FollandMoments) (T U : Mat3)
(hT : T.trace = 0) (hU : U.trace = 0) :
(∫ q, squareNorm (normalDeviation T U q) ∂sphereMeasure) =
(3 / 2) * (Geometry.normalizedBlockNorm T ^ 2 +
Geometry.normalizedBlockNorm U ^ 2) := by
rw [integral_squareNorm_normalDeviation h T U hT hU,
Geometry.normalizedBlockNorm_sq, Geometry.normalizedBlockNorm_sq]
unfold squareNorm
ring
end
end PositiveEinsteinFour.Angular
end OAI
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