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695bd70 | 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 | import OAI.Geometry.Relativity.CKS.InducedSphereCalculus
namespace OAI
noncomputable section
namespace CKSSphericalChart
noncomputable section
open Set Filter Finset CKSCalculus CKSRealizedRound
open CKSInducedSphere (E Ix)
open scoped Topology ContDiff
lemma D_line {F : Point → ℝ} {x : Point} (hF : DifferentiableAt ℝ F x) (v : Point) :
D v F x = deriv (fun s : ℝ => F (x + s • v)) 0 := by
have hd : HasDerivAt (fun s : ℝ => x + s • v) v 0 := by
convert! ((hasDerivAt_id (0:ℝ)).smul_const v).const_add x using 1
simp
have hf : HasFDerivAt F (fderiv ℝ F x) (x + (0:ℝ) • v) := by simpa using hF.hasFDerivAt
have hh := hf.comp_hasDerivAt 0 hd
simpa only [zero_smul,add_zero,D,Function.comp_def] using hh.deriv.symm
lemma D_line_congr {F G : Point → ℝ} {x : Point}
(hF : DifferentiableAt ℝ F x) (hG : DifferentiableAt ℝ G x) (v : Point)
(he : ∀ s : ℝ, F (x+s•v) = G (x+s•v)) : D v F x = D v G x := by
rw [D_line hF,D_line hG]
congr 1
funext s
exact he s
lemma angular_line_radial (x : Point) (k : Ix) (hk : k ≠ 0) (s : ℝ) :
(x+s•basis k) 0 = x 0 := by
simp [basis,Ne.symm hk]
lemma angular_first_congr_leaf {F G : Point → ℝ} {x : Point}
(hF : DifferentiableAt ℝ F x) (hG : DifferentiableAt ℝ G x)
(he : ∀ y : Point, y 0 = x 0 → F y = G y) (k : Ix) (hk : k ≠ 0) :
D (basis k) F x = D (basis k) G x := by
apply D_line_congr hF hG
intro s
exact he _ (angular_line_radial x k hk s)
lemma angular_second_congr_leaf {F G : Point → ℝ} {x : Point}
(hF : ∀ y : Point, y 0 = x 0 → ContDiffAt ℝ 2 F y)
(hG : ∀ y : Point, y 0 = x 0 → ContDiffAt ℝ 2 G y)
(he : ∀ y : Point, y 0 = x 0 → F y = G y)
(i j : Ix) (hi : i ≠ 0) (hj : j ≠ 0) :
D (basis j) (D (basis i) F) x = D (basis j) (D (basis i) G) x := by
have hFd := (CKSCalculus.contDiffAt_D (hF x rfl) (m := 1) (by norm_num) (basis i)).differentiableAt (by norm_num)
have hGd := (CKSCalculus.contDiffAt_D (hG x rfl) (m := 1) (by norm_num) (basis i)).differentiableAt (by norm_num)
apply D_line_congr hFd hGd
intro s
have hy := angular_line_radial x j hj s
apply angular_first_congr_leaf ((hF _ hy).differentiableAt (by norm_num))
((hG _ hy).differentiableAt (by norm_num)) _ i hi
intro z hz
exact he z (hz.trans hy)
end
end CKSSphericalChart
end
end OAI
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