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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import OAI.Geometry.IsometricImmersion.Metric
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
namespace OAI
noncomputable section
open Set Filter Function TopologicalSpace
open scoped ContDiff Topology Manifold Matrix
namespace SmoothLocal.Geometry
section Extension
variable {E : Type*}
def openDomainExtend (U : Opens Coord) (F : U → E) (outside : E) : Coord → E :=
Function.extend Subtype.val F (fun _ => outside)
@[simp] theorem openDomainExtend_apply (U : Opens Coord) (F : U → E)
(outside : E) (p : U) : openDomainExtend U F outside p = F p :=
Subtype.val_injective.extend_apply F (fun _ => outside) p
theorem openDomainExtend_restrict (U : Opens Coord) (F : U → E)
(outside : E) : (fun p : U => openDomainExtend U F outside p) = F := by
funext p
exact openDomainExtend_apply U F outside p
variable [NormedAddCommGroup E] [NormedSpace ℝ E]
theorem contMDiff_restrict_iff (U : Opens Coord) (F : Coord → E)
(n : ℕ∞ω) :
ContMDiff 𝓘(ℝ, Coord) 𝓘(ℝ, E) n (fun p : U => F p) ↔
ContDiffOn ℝ n F (U : Set Coord) := by
constructor
· intro h p hp
have hs := contMDiffAt_subtype_iff.mp (h ⟨p, hp⟩)
exact hs.contDiffAt.contDiffWithinAt
· intro h p
apply contMDiffAt_subtype_iff.mpr
exact ((h p p.property).contDiffAt (U.isOpen.mem_nhds p.property)).contMDiffAt
theorem openDomainExtend_contDiffOn (U : Opens Coord) (F : U → E)
(outside : E) {n : ℕ∞ω}
(hF : ContMDiff 𝓘(ℝ, Coord) 𝓘(ℝ, E) n F) :
ContDiffOn ℝ n (openDomainExtend U F outside) (U : Set Coord) := by
apply (contMDiff_restrict_iff U _ n).mp
rw [openDomainExtend_restrict]
exact hF
theorem mfderiv_restrict_eq_fderiv (U : Opens Coord) (F : Coord → E)
(p : U) (hF : ContDiffAt ℝ ∞ F (p : Coord)) :
mfderiv 𝓘(ℝ, Coord) 𝓘(ℝ, E) (fun q : U => F q) p =
fderiv ℝ F (p : Coord) := by
have hs : ContMDiffAt 𝓘(ℝ, Coord) 𝓘(ℝ, E) ∞
(fun q : U => F q) p :=
contMDiffAt_subtype_iff.mpr hF.contMDiffAt
have hd := hs.mdifferentiableAt (by simp : (∞ : ℕ∞ω) ≠ 0)
have hc : (fun y : Coord => ((chartAt Coord p).symm y : Coord))
=ᶠ[𝓝 (p : Coord)] (fun y : Coord => y) := by
exact (U.chartAt_subtype_val_symm_eventuallyEq (H := Coord) (x := p)).symm
have he : writtenInExtChartAt 𝓘(ℝ, Coord) 𝓘(ℝ, E) p
(fun q : U => F q) =ᶠ[𝓝 (p : Coord)] F := by
exact hc.fun_comp F
rw [hd.mfderiv]
change fderivWithin ℝ
(writtenInExtChartAt 𝓘(ℝ, Coord) 𝓘(ℝ, E) p (fun q : U => F q))
(range 𝓘(ℝ, Coord)) (p : Coord) = _
rw [ModelWithCorners.range_eq_univ, fderivWithin_univ]
exact he.fderiv_eq
theorem fderiv_openDomainExtend_eq_mfderiv (U : Opens Coord) (F : U → E)
(outside : E) (hF : ContMDiff 𝓘(ℝ, Coord) 𝓘(ℝ, E) ∞ F) (p : U) :
fderiv ℝ (openDomainExtend U F outside) (p : Coord) =
mfderiv 𝓘(ℝ, Coord) 𝓘(ℝ, E) F p := by
have hd := ((openDomainExtend_contDiffOn U F outside hF) p p.property).contDiffAt
(U.isOpen.mem_nhds p.property)
have he := mfderiv_restrict_eq_fderiv U (openDomainExtend U F outside) p hd
rw [openDomainExtend_restrict] at he
exact he.symm
end Extension
def LocalIsometric (g : MetricField) (U : Opens Coord) (F : U → Ambient) : Prop :=
ContMDiff 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) ∞ F ∧
∀ p : U, ∀ v w : Coord,
(inner ℝ : Ambient → Ambient → ℝ) (mfderiv 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) F p v)
(mfderiv 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) F p w) =
v ⬝ᵥ (g p *ᵥ w)
theorem localIsometric_restrict_iff (g : MetricField) (U : Opens Coord)
(F : Coord → Ambient) :
LocalIsometric g U (fun p : U => F p) ↔ IsometricOn g F (U : Set Coord) := by
constructor
· rintro ⟨hF, he⟩
have hc := (contMDiff_restrict_iff U F ∞).mp hF
refine ⟨hc, ?_⟩
intro p hp v w
have hd := mfderiv_restrict_eq_fderiv U F ⟨p, hp⟩
((hc p hp).contDiffAt (U.isOpen.mem_nhds hp))
have hinner := he ⟨p, hp⟩ v w
rw [hd] at hinner
exact hinner
· rintro ⟨hF, he⟩
refine ⟨(contMDiff_restrict_iff U F ∞).mpr hF, ?_⟩
intro p v w
rw [mfderiv_restrict_eq_fderiv U F p
((hF p p.property).contDiffAt (U.isOpen.mem_nhds p.property))]
exact he p p.property v w
theorem localIsometric_extend_iff (g : MetricField) (U : Opens Coord)
(F : U → Ambient) (outside : Ambient) :
LocalIsometric g U F ↔ IsometricOn g (openDomainExtend U F outside) U := by
rw [← localIsometric_restrict_iff, openDomainExtend_restrict]
theorem localIsometric_injective_differential {g : MetricField} {U : Opens Coord}
{F : U → Ambient} (hg : SmoothPositiveOn g U) (hF : LocalIsometric g U F)
(p : U) : Function.Injective (mfderiv 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) F p) := by
rw [← fderiv_openDomainExtend_eq_mfderiv U F 0 hF.1 p]
exact isometricOn_injective_differential hg
((localIsometric_extend_iff g U F 0).mp hF) p.property
def LocalSmoothPositive (U : Opens Coord)
(g : U → Matrix (Fin 2) (Fin 2) ℝ) : Prop :=
(∀ i j, ContMDiff 𝓘(ℝ, Coord) 𝓘(ℝ, ℝ) ∞ (fun p => g p i j)) ∧
∀ p, (g p).PosDef
theorem localSmoothPositive_restrict_iff (U : Opens Coord) (g : MetricField) :
LocalSmoothPositive U (fun p => g p) ↔ SmoothPositiveOn g U := by
constructor
· rintro ⟨hs, hp⟩
exact ⟨fun i j => (contMDiff_restrict_iff U (fun p => g p i j) ∞).mp (hs i j),
fun p h => hp ⟨p, h⟩⟩
· rintro ⟨hs, hp⟩
exact ⟨fun i j => (contMDiff_restrict_iff U (fun p => g p i j) ∞).mpr (hs i j),
fun p => hp p p.property⟩
theorem localSmoothPositive_extend_iff (U : Opens Coord)
(g : U → Matrix (Fin 2) (Fin 2) ℝ) (outside : Matrix (Fin 2) (Fin 2) ℝ) :
LocalSmoothPositive U g ↔ SmoothPositiveOn
(Function.extend Subtype.val g (fun _ => outside)) U := by
rw [← localSmoothPositive_restrict_iff]
have he : (fun p : U => Function.extend Subtype.val g (fun _ => outside) p) = g := by
funext p
exact Subtype.val_injective.extend_apply g (fun _ => outside) p
rw [he]
end SmoothLocal.Geometry
end
end OAI
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