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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import OAI.Geometry.IsometricImmersion.Metric
import OAI.Geometry.IsometricImmersion.Energy.HeightEnergy
import Mathlib.Topology.Algebra.Support
import Mathlib.Analysis.Matrix.Normed
namespace OAI
noncomputable section
open scoped ContDiff Topology BigOperators Matrix
open scoped Matrix.Norms.Elementwise
open Filter
namespace SmoothLocal.Geometry
theorem coordPartial_eq_of_eventuallyEq
{V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
{f h : Coord → V} {p : Coord} (heq : f =ᶠ[𝓝 p] h) (i : Fin 2) :
coordPartial i f p = coordPartial i h p := by
unfold coordPartial
rw [heq.fderiv_eq]
theorem coordPartial_eventuallyEq
{V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
{f h : Coord → V} {p : Coord} (heq : f =ᶠ[𝓝 p] h) (i : Fin 2) :
coordPartial i f =ᶠ[𝓝 p] coordPartial i h := by
filter_upwards [heq.eventuallyEq_nhds] with q hq
exact coordPartial_eq_of_eventuallyEq hq i
def iteratedCoordPartial {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] :
List (Fin 2) → (Coord → V) → Coord → V
| [], f => f
| i :: is, f => coordPartial i (iteratedCoordPartial is f)
theorem iteratedCoordPartial_eventuallyEq
{V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
{f h : Coord → V} {p : Coord} (heq : f =ᶠ[𝓝 p] h) (dirs : List (Fin 2)) :
iteratedCoordPartial dirs f =ᶠ[𝓝 p] iteratedCoordPartial dirs h := by
induction dirs with
| nil => exact heq
| cons i is ih => exact coordPartial_eventuallyEq ih i
theorem iteratedCoordPartial_zero
{V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
(dirs : List (Fin 2)) : iteratedCoordPartial dirs (0 : Coord → V) = 0 := by
induction dirs with
| nil => rfl
| cons i is ih =>
change coordPartial i (iteratedCoordPartial is (0 : Coord → V)) = 0
rw [ih]
funext p
change fderiv ℝ (fun _ : Coord => (0 : V)) p (Pi.single i 1) = 0
simp
theorem eventuallyEq_zero_of_not_mem_tsupport
{V : Type*} [Zero V] (f : Coord → V) {p : Coord} (hp : p ∉ tsupport f) :
f =ᶠ[𝓝 p] 0 := by
filter_upwards [(isClosed_tsupport f).isOpen_compl.mem_nhds hp] with q hq
change f q = 0
by_contra hne
exact hq (subset_tsupport f hne)
theorem iteratedCoordPartial_zero_of_not_mem_tsupport
{V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
(f : Coord → V) {p : Coord} (hp : p ∉ tsupport f) (dirs : List (Fin 2)) :
iteratedCoordPartial dirs f p = 0 := by
have heq := iteratedCoordPartial_eventuallyEq
(eventuallyEq_zero_of_not_mem_tsupport f hp) dirs
rw [heq.self_of_nhds, iteratedCoordPartial_zero]
rfl
theorem metric_coeff_eventuallyEq {g h : MetricField} {p : Coord}
(heq : g =ᶠ[𝓝 p] h) (i j : Fin 2) :
(fun q => g q i j) =ᶠ[𝓝 p] (fun q => h q i j) :=
heq.mono fun _ hq => congrFun (congrFun hq i) j
theorem christoffel_eq_of_eventuallyEq {g h : MetricField} {p : Coord}
(heq : g =ᶠ[𝓝 p] h) (k i j : Fin 2) :
christoffel g k i j p = christoffel h k i j p := by
have hval := heq.self_of_nhds
have hd (a b c : Fin 2) : coordPartial a (fun q => g q b c) p =
coordPartial a (fun q => h q b c) p :=
coordPartial_eq_of_eventuallyEq (metric_coeff_eventuallyEq heq b c) a
simp only [christoffel, inverseMetric, hval, hd]
theorem christoffel_eventuallyEq {g h : MetricField} {p : Coord}
(heq : g =ᶠ[𝓝 p] h) (k i j : Fin 2) :
christoffel g k i j =ᶠ[𝓝 p] christoffel h k i j :=
heq.eventuallyEq_nhds.mono fun _ hq => christoffel_eq_of_eventuallyEq hq k i j
theorem riemann_eq_of_eventuallyEq {g h : MetricField} {p : Coord}
(heq : g =ᶠ[𝓝 p] h) (l k i j : Fin 2) :
riemann g l k i j p = riemann h l k i j p := by
have hΓ := christoffel_eq_of_eventuallyEq heq
have hd (a b c d : Fin 2) : coordPartial a (christoffel g b c d) p =
coordPartial a (christoffel h b c d) p :=
coordPartial_eq_of_eventuallyEq (christoffel_eventuallyEq heq b c d) a
simp only [riemann, hΓ, hd]
theorem gaussianCurvature_eq_of_eventuallyEq {g h : MetricField} {p : Coord}
(heq : g =ᶠ[𝓝 p] h) : gaussianCurvature g p = gaussianCurvature h p := by
simp only [gaussianCurvature, heq.self_of_nhds, riemann_eq_of_eventuallyEq heq]
theorem covHessian_eq_of_metric_eventuallyEq {g h : MetricField} {p : Coord}
(heq : g =ᶠ[𝓝 p] h) (z : Coord → ℝ) : covHessian g z p = covHessian h z p := by
ext i j
simp only [covHessian, christoffel_eq_of_eventuallyEq heq]
theorem heightEnergy_eq_of_metric_eventuallyEq {g h : MetricField} {p : Coord}
(heq : g =ᶠ[𝓝 p] h) (z : Coord → ℝ) : heightEnergy g z p = heightEnergy h z p := by
simp only [heightEnergy, covectorNormSq, inverseMetric, heq.self_of_nhds]
theorem metric_add_eventuallyEq_of_not_mem_tsupport (g η : MetricField)
{p : Coord} (hp : p ∉ tsupport η) : g + η =ᶠ[𝓝 p] g := by
filter_upwards [eventuallyEq_zero_of_not_mem_tsupport η hp] with q hq
change g q + η q = g q
simp only [hq, Pi.zero_apply, add_zero]
theorem metric_add_all_coordinate_jets (g η : MetricField)
{p : Coord} (hp : p ∉ tsupport η) (dirs : List (Fin 2)) :
iteratedCoordPartial dirs (g + η) p = iteratedCoordPartial dirs g p :=
(iteratedCoordPartial_eventuallyEq
(metric_add_eventuallyEq_of_not_mem_tsupport g η hp) dirs).self_of_nhds
theorem gaussianCurvature_add_eq_off_tsupport (g η : MetricField)
{p : Coord} (hp : p ∉ tsupport η) :
gaussianCurvature (g + η) p = gaussianCurvature g p :=
gaussianCurvature_eq_of_eventuallyEq (metric_add_eventuallyEq_of_not_mem_tsupport g η hp)
end SmoothLocal.Geometry
end
end OAI
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