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8e4d533 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 | import OAI.Geometry.ComplexBundle.DirectSum
import Mathlib.Analysis.Complex.Basic
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
import Mathlib.Geometry.Manifold.VectorBundle.Basic
namespace OAI
noncomputable section
/-! Real models of complex bundles and smooth finite line sums. -/
section
open Bundle Topology
open scoped Manifold ContDiff
namespace Kaplansky
section
variable {B F J : Type*} [TopologicalSpace B]
[NormedAddCommGroup F] [NormedSpace ℂ F] (C : VectorBundleCore ℂ B F J)
/-- Restrict scalars while retaining the original complex transition maps
as functions, and hence the original bundle atlas and topology. -/
def complexCoreReal : VectorBundleCore ℝ B F J where
baseSet := C.baseSet
isOpen_baseSet := C.isOpen_baseSet
indexAt := C.indexAt
mem_baseSet_at := C.mem_baseSet_at
coordChange i j x := (C.coordChange i j x).restrictScalars ℝ
coordChange_self := C.coordChange_self
continuousOn_coordChange i j :=
(ContinuousLinearMap.restrictScalarsL ℂ F F ℝ ℝ).continuous.comp_continuousOn
(C.continuousOn_coordChange i j)
coordChange_comp := C.coordChange_comp
theorem complexCoreReal_toFiberBundleCore : (complexCoreReal C).toFiberBundleCore = C.toFiberBundleCore := rfl
instance complexCoreFiberRealModule (x : B) : Module ℝ (C.Fiber x) := inferInstanceAs (Module ℝ F)
instance complexCoreRealFiberComplexModule (x : B) : Module ℂ ((complexCoreReal C).Fiber x) :=
inferInstanceAs (Module ℂ F)
instance complexCoreFiberRealVectorBundle : VectorBundle ℝ F C.Fiber :=
inferInstanceAs (VectorBundle ℝ F (complexCoreReal C).Fiber)
instance complexCoreRealFiberComplexVectorBundle : VectorBundle ℂ F (complexCoreReal C).Fiber :=
inferInstanceAs (VectorBundle ℂ F C.Fiber)
/-- Real-smooth complex transition functions produce an ordinary real-smooth
complex bundle, with no separate realization or atlas-compatibility assumption. -/
theorem complexCoreReal_isContMDiff
{X H : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] [TopologicalSpace H]
(I : ModelWithCorners ℝ X H) [ChartedSpace H B]
(hC : ∀ a b, ContMDiffOn I 𝓘(ℝ,F →L[ℂ] F) ∞ (C.coordChange a b)
(C.baseSet a ∩ C.baseSet b)) : (complexCoreReal C).IsContMDiff I ∞ where
contMDiffOn_coordChange a b := by
intro x hx
exact (ContinuousLinearMap.restrictScalarsL ℂ F F ℝ ℝ).contDiff.contDiffAt.contMDiffAt.comp_contMDiffWithinAt x
(hC a b x hx)
theorem complexCore_contMDiffVectorBundle
{X H : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] [TopologicalSpace H]
(I : ModelWithCorners ℝ X H) [ChartedSpace H B]
(hC : ∀ a b, ContMDiffOn I 𝓘(ℝ,F →L[ℂ] F) ∞ (C.coordChange a b)
(C.baseSet a ∩ C.baseSet b)) : ContMDiffVectorBundle ∞ F C.Fiber I := by
let : (complexCoreReal C).IsContMDiff I ∞ := complexCoreReal_isContMDiff C I hC
exact inferInstanceAs (ContMDiffVectorBundle ∞ F (complexCoreReal C).Fiber I)
end
variable {S : Type*} [Fintype S]
def lineDiagonal (c : S → ℂ) : EuclideanSpace ℂ S →L[ℂ] EuclideanSpace ℂ S :=
(EuclideanSpace.equiv S ℂ).symm.toContinuousLinearMap.comp
((ContinuousLinearMap.pi (fun s => c s • (ContinuousLinearMap.proj s : (S → ℂ) →L[ℂ] ℂ))).comp
(EuclideanSpace.equiv S ℂ).toContinuousLinearMap)
def lineDiagonalLinear : (S → ℂ) →ₗ[ℂ] (EuclideanSpace ℂ S →L[ℂ] EuclideanSpace ℂ S) where
toFun := lineDiagonal
map_add' c d := by ext v s; change (c s+d s)*v s=c s*v s+d s*v s; ring
map_smul' z c := by ext v s; change (z*c s)*v s=z*(c s*v s); ring
variable {B : Type*} [TopologicalSpace B] {J : S → Type*}
(C : ∀ s, VectorBundleCore ℂ B ℂ (J s))
omit [Fintype S] in
theorem lineSumTransition_eq_diagonal (a b : ∀ s, J s) (x : B) :
lineSumTransition C a b x = lineDiagonal (fun s => (C s).coordChange (a s) (b s) x 1) := by
ext v s
change (C s).coordChange (a s) (b s) x (v s) =
((C s).coordChange (a s) (b s) x 1)*(v s)
have h := map_smul ((C s).coordChange (a s) (b s) x) (v s) (1 : ℂ)
simpa only [smul_eq_mul,mul_one,one_mul,mul_comm] using h
theorem finiteLineSumCore_transition_smooth
{X H : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] [TopologicalSpace H]
(I : ModelWithCorners ℝ X H) [ChartedSpace H B]
(hC : ∀ s a b, ContMDiffOn I 𝓘(ℝ,ℂ →L[ℂ] ℂ) ∞ ((C s).coordChange a b)
((C s).baseSet a ∩ (C s).baseSet b)) (a b : ∀ s, J s) :
ContMDiffOn I 𝓘(ℝ,EuclideanSpace ℂ S →L[ℂ] EuclideanSpace ℂ S) ∞
((finiteLineSumCore C).coordChange a b)
((finiteLineSumCore C).baseSet a ∩ (finiteLineSumCore C).baseSet b) := by
have hc : ContMDiffOn I 𝓘(ℝ,S → ℂ) ∞
(fun x s => (C s).coordChange (a s) (b s) x 1)
((finiteLineSumCore C).baseSet a ∩ (finiteLineSumCore C).baseSet b) := by
apply contMDiffOn_pi_space.mpr
intro s x hx
exact ((ContinuousLinearMap.apply ℂ ℂ (1 : ℂ)).restrictScalars ℝ).contDiff.contDiffAt.contMDiffAt.comp_contMDiffWithinAt x
((hC s (a s) (b s)).mono (fun y (hy : y ∈ (finiteLineSumCore C).baseSet a ∩ (finiteLineSumCore C).baseSet b) => ⟨hy.1 s,hy.2 s⟩) x hx)
let hd := (lineDiagonalLinear (S := S)).toContinuousLinearMap.restrictScalars ℝ
intro x hx
have h := hd.contDiff.contDiffAt.contMDiffAt.comp_contMDiffWithinAt x (hc x hx)
exact h.congr (fun y _ => by simpa [hd, finiteLineSumCore, lineDiagonalLinear] using lineSumTransition_eq_diagonal C a b y) (by simpa [hd, finiteLineSumCore, lineDiagonalLinear] using lineSumTransition_eq_diagonal C a b x)
end Kaplansky
end
open Bundle Topology
open scoped Manifold ContDiff
namespace Kaplansky
variable {B F G J : Type*} [TopologicalSpace B]
[NormedAddCommGroup F] [NormedSpace ℂ F]
[NormedAddCommGroup G] [NormedSpace ℂ G]
(C : VectorBundleCore ℂ B F J) (e : F ≃L[ℂ] G)
/-- A fixed linear change of model coordinates; the gluing is conjugated. -/
def changeModelCore : VectorBundleCore ℂ B G J where
baseSet := C.baseSet
isOpen_baseSet := C.isOpen_baseSet
indexAt := C.indexAt
mem_baseSet_at := C.mem_baseSet_at
coordChange i j x := e.toContinuousLinearMap.comp ((C.coordChange i j x).comp e.symm.toContinuousLinearMap)
coordChange_self i x hx v := by simp [C.coordChange_self i x hx]
continuousOn_coordChange i j := continuousOn_const.clm_comp ((C.continuousOn_coordChange i j).clm_comp continuousOn_const)
coordChange_comp i j k x hx v := by
simpa only [ContinuousLinearMap.comp_apply,ContinuousLinearEquiv.coe_coe,ContinuousLinearEquiv.symm_apply_apply] using
congrArg e (C.coordChange_comp i j k x hx (e.symm v))
/-- The coordinate change is a homeomorphism of actual total spaces. -/
def changeModelCoreTotalHomeomorph : TotalSpace F C.Fiber ≃ₜ TotalSpace G (changeModelCore C e).Fiber where
toFun z := ⟨z.proj,e z.2⟩
invFun z := ⟨z.proj,e.symm z.2⟩
left_inv z := by
obtain ⟨x, v⟩ := z
exact congrArg (fun w => TotalSpace.mk' F x w) (e.symm_apply_apply v)
right_inv z := by
obtain ⟨x, v⟩ := z
exact congrArg (fun w => TotalSpace.mk' G x w) (e.apply_symm_apply v)
continuous_toFun := by
apply continuous_iff_continuousAt.mpr
intro z
rw [FiberBundle.continuousAt_totalSpace]
refine ⟨(FiberBundle.continuous_proj F C.Fiber).continuousAt,?_⟩
have h := ((C.localTriv (C.indexAt z.proj)).continuousAt
((C.localTriv _).mem_source.mpr (C.mem_baseSet_at z.proj))).snd
change ContinuousAt (fun x : TotalSpace F C.Fiber =>
e (C.coordChange (C.indexAt x.proj) (C.indexAt z.proj) x.proj (e.symm (e x.2)))) z
change ContinuousAt (fun x : TotalSpace F C.Fiber =>
C.coordChange (C.indexAt x.proj) (C.indexAt z.proj) x.proj x.2) z at h
have he (x : TotalSpace F C.Fiber) : e.symm (e x.2) = x.2 := e.symm_apply_apply x.2
simpa only [Function.comp_def, he] using e.continuous.continuousAt.comp h
continuous_invFun := by
apply continuous_iff_continuousAt.mpr
intro z
rw [FiberBundle.continuousAt_totalSpace]
refine ⟨(FiberBundle.continuous_proj G (changeModelCore C e).Fiber).continuousAt,?_⟩
have h := (((changeModelCore C e).localTriv (C.indexAt z.proj)).continuousAt
(((changeModelCore C e).localTriv _).mem_source.mpr (C.mem_baseSet_at z.proj))).snd
change ContinuousAt (fun x : TotalSpace G (changeModelCore C e).Fiber =>
C.coordChange (C.indexAt x.proj) (C.indexAt z.proj) x.proj (e.symm x.2)) z
change ContinuousAt (fun x : TotalSpace G (changeModelCore C e).Fiber =>
e (C.coordChange (C.indexAt x.proj) (C.indexAt z.proj) x.proj (e.symm x.2))) z at h
simpa only [Function.comp_def,ContinuousLinearEquiv.symm_apply_apply] using e.symm.continuous.continuousAt.comp h
theorem changeModelCore_transition_smooth
{X H : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] [TopologicalSpace H]
(I : ModelWithCorners ℝ X H) [ChartedSpace H B]
(hC : ∀ a b, ContMDiffOn I 𝓘(ℝ,F →L[ℂ] F) ∞ (C.coordChange a b)
(C.baseSet a ∩ C.baseSet b)) (a b : J) :
ContMDiffOn I 𝓘(ℝ,G →L[ℂ] G) ∞ ((changeModelCore C e).coordChange a b)
((changeModelCore C e).baseSet a ∩ (changeModelCore C e).baseSet b) := by
let L : (F →L[ℂ] F) →L[ℂ] (G →L[ℂ] G) :=
(ContinuousLinearMap.compL ℂ G F G e.toContinuousLinearMap).comp
((ContinuousLinearMap.compL ℂ G F F).flip e.symm.toContinuousLinearMap)
intro x hx
exact (L.restrictScalars ℝ).contDiff.contDiffAt.contMDiffAt.comp_contMDiffWithinAt x (hC a b x hx)
instance changeModelFiberNorm (x : B) : NormedAddCommGroup ((changeModelCore C e).Fiber x) :=
inferInstanceAs (NormedAddCommGroup G)
instance changeModelFiberNormedSpace (x : B) : NormedSpace ℂ ((changeModelCore C e).Fiber x) :=
inferInstanceAs (NormedSpace ℂ G)
end Kaplansky
end
end OAI
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