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59d1c80 | 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 | import OAI.Geometry.IsometricImmersion.Immersions.NormalField
import Mathlib.Tactic.Linarith
namespace OAI
noncomputable section
open scoped ContDiff Topology BigOperators Matrix
open Filter
namespace SmoothLocal.Geometry
variable {g : MetricField} {F n : Coord → Ambient} {U : Set Coord} {p : Coord}
theorem normalField_derivative_inner_self (hn : ContDiffOn ℝ ∞ n U)
(hunit : ∀ q ∈ U, inner ℝ (n q) (n q) = 1)
(hU : IsOpen U) (hp : p ∈ U) (i : Fin 2) :
inner ℝ (coordPartial i n p) (n p) = 0 := by
have hd : DifferentiableAt ℝ n p :=
(((hn p hp).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp))
have heq : (fun q => inner ℝ (n q) (n q)) =ᶠ[𝓝 p]
(fun _ : Coord => (1 : ℝ)) := by
filter_upwards [hU.mem_nhds hp] with q hq
exact hunit q hq
have heq' := congrArg (fun L : Coord →L[ℝ] ℝ => L (Pi.single i 1)) heq.fderiv_eq
rw [fderiv_inner_apply ℝ hd hd] at heq'
simp only [fderiv_const_apply, zero_apply] at heq'
change inner ℝ (n p) (coordPartial i n p) +
inner ℝ (coordPartial i n p) (n p) = 0 at heq'
rw [← real_inner_comm (n p) (coordPartial i n p)] at heq'
linarith
theorem normalField_derivative_inner_tangent (hF : ContDiffOn ℝ ∞ F U)
(hn : ContDiffOn ℝ ∞ n U) (hnormal : ∀ q ∈ U, IsUnitNormalAt F (n q) q)
(hU : IsOpen U) (hp : p ∈ U) (i j : Fin 2) :
inner ℝ (coordPartial i n p) (coordPartial j F p) =
-secondFundamental F (n p) p i j := by
have hdN : DifferentiableAt ℝ n p :=
(((hn p hp).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp))
have hdT : DifferentiableAt ℝ (coordPartial j F) p :=
((((partial_contDiffOn hF hU j) p hp).contDiffAt
(hU.mem_nhds hp)).differentiableAt (by simp))
have heq : (fun q => inner ℝ (n q) (coordPartial j F q)) =ᶠ[𝓝 p]
(fun _ : Coord => (0 : ℝ)) := by
filter_upwards [hU.mem_nhds hp] with q hq
rw [real_inner_comm]
exact (hnormal q hq).2 (Pi.single j 1)
have heq' := congrArg (fun L : Coord →L[ℝ] ℝ => L (Pi.single i 1)) heq.fderiv_eq
rw [fderiv_inner_apply ℝ hdN hdT] at heq'
simp only [fderiv_const_apply, zero_apply] at heq'
change inner ℝ (n p) (coordPartial i (coordPartial j F) p) +
inner ℝ (coordPartial i n p) (coordPartial j F p) = 0 at heq'
rw [← real_inner_comm (n p) (coordPartial i (coordPartial j F) p)] at heq'
change secondFundamental F (n p) p i j +
inner ℝ (coordPartial i n p) (coordPartial j F p) = 0 at heq'
linarith
theorem coordPartial_normalField_eq_zero_of_secondFundamental_eq_zero
(hg : SmoothPositiveOn g U) (hF : IsometricOn g F U)
(hn : ContDiffOn ℝ ∞ n U) (hnormal : ∀ q ∈ U, IsUnitNormalAt F (n q) q)
(hU : IsOpen U) (hp : p ∈ U)
(hII : secondFundamental F (n p) p = 0) (i : Fin 2) :
coordPartial i n p = 0 := by
have hnt (j : Fin 2) : inner ℝ (coordPartial j F p) (n p) = 0 :=
(hnormal p hp).2 (Pi.single j 1)
have hwt (j : Fin 2) : inner ℝ (coordPartial j F p) (coordPartial i n p) = 0 := by
rw [real_inner_comm, normalField_derivative_inner_tangent hF.1 hn hnormal hU hp i j]
rw [hII]
simp
have hnw : inner ℝ (n p) (coordPartial i n p) = 0 := by
rw [real_inner_comm]
exact normalField_derivative_inner_self hn (fun q hq => (hnormal q hq).1) hU hp i
have hspan := normal_eq_inner_smul_of_independent_tangents
(fun j => coordPartial j F p) (isometric_tangents_independent hg hF hp)
(n p) (coordPartial i n p) (hnormal p hp).1 hnt hwt
simpa only [hnw, zero_smul] using hspan
theorem fderiv_normalField_eq_zero_of_secondFundamental_eq_zero
(hg : SmoothPositiveOn g U) (hF : IsometricOn g F U)
(hn : ContDiffOn ℝ ∞ n U) (hnormal : ∀ q ∈ U, IsUnitNormalAt F (n q) q)
(hU : IsOpen U) (hp : p ∈ U)
(hII : secondFundamental F (n p) p = 0) : fderiv ℝ n p = 0 := by
apply ContinuousLinearMap.ext
intro v
rw [← Finset.univ_sum_single v, map_sum]
change (∑ i : Fin 2, fderiv ℝ n p (Pi.single i (v i))) = 0
apply Finset.sum_eq_zero
intro i _
have hsingle : Pi.single i (v i) = v i • Pi.single i (1 : ℝ) := by
simpa only [smul_eq_mul, mul_one] using Pi.single_smul' i (v i) (1 : ℝ)
rw [hsingle, map_smul]
change v i • coordPartial i n p = 0
rw [coordPartial_normalField_eq_zero_of_secondFundamental_eq_zero
hg hF hn hnormal hU hp hII i, smul_zero]
theorem unitNormalField_derivative_inner_self
(hg : SmoothPositiveOn g U) (hF : IsometricOn g F U)
(hU : IsOpen U) (hp : p ∈ U) (i : Fin 2) :
inner ℝ (coordPartial i (unitNormalField F) p) (unitNormalField F p) = 0 :=
normalField_derivative_inner_self (unitNormalField_contDiffOn hg hF hU)
(fun _ hq => unitNormalField_inner_self hg hF hq) hU hp i
theorem unitNormalField_derivative_inner_tangent
(hg : SmoothPositiveOn g U) (hF : IsometricOn g F U)
(hU : IsOpen U) (hp : p ∈ U) (i j : Fin 2) :
inner ℝ (coordPartial i (unitNormalField F) p) (coordPartial j F p) =
-secondFundamental F (unitNormalField F p) p i j :=
normalField_derivative_inner_tangent hF.1 (unitNormalField_contDiffOn hg hF hU)
(fun _ hq => unitNormalField_isUnitNormalAt hg hF hq) hU hp i j
theorem unitNormalField_fderiv_eq_zero_of_secondFundamental_eq_zero
(hg : SmoothPositiveOn g U) (hF : IsometricOn g F U)
(hU : IsOpen U) (hp : p ∈ U)
(hII : secondFundamental F (unitNormalField F p) p = 0) :
fderiv ℝ (unitNormalField F) p = 0 :=
fderiv_normalField_eq_zero_of_secondFundamental_eq_zero hg hF
(unitNormalField_contDiffOn hg hF hU)
(fun _ hq => unitNormalField_isUnitNormalAt hg hF hq) hU hp hII
end SmoothLocal.Geometry
end
end OAI
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