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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 | import OAI.Geometry.ComplexBundle.Frames
import OAI.Geometry.ComplexBundle.SmoothLine
namespace OAI
noncomputable section
/-! Finite-coordinate smooth bundles and their metrics. -/
section
open Bundle Topology
open scoped Manifold ContDiff
namespace Kaplansky
open ComplexGrassmann
variable {S ι : Type*} [Fintype S] [Fintype ι]
def finiteCoordinateEquiv (d : ℕ) (hd : Fintype.card S=d) :
EuclideanSpace ℂ S ≃L[ℂ] (Fin d → ℂ) :=
(LinearEquiv.ofFinrankEq (R := ℂ) (EuclideanSpace ℂ S) (Fin d → ℂ) (by simpa using hd)).toContinuousLinearEquiv
variable (K : ι → ℕ) (label : S → Option ι) (d : ℕ) (hd : Fintype.card S=d)
def finiteCoordinateCore := changeModelCore (smoothLineSumCore K label) (finiteCoordinateEquiv d hd)
abbrev FiniteCoordinateBundle := (finiteCoordinateCore K label d hd).Fiber
instance finiteCoordinateBundle_norm (P : ProjectiveProduct K) :
NormedAddCommGroup (FiniteCoordinateBundle K label d hd P) := changeModelFiberNorm _ _ P
instance finiteCoordinateBundle_normedSpace (P : ProjectiveProduct K) :
NormedSpace ℂ (FiniteCoordinateBundle K label d hd P) := changeModelFiberNormedSpace _ _ P
instance finiteCoordinateBundle_smooth : ContMDiffVectorBundle ∞ (Fin d → ℂ)
(FiniteCoordinateBundle K label d hd) 𝓘(ℝ,ProjectiveProductModel K) :=
complexCore_contMDiffVectorBundle _ _ (changeModelCore_transition_smooth _ _ _
(finiteLineSumCore_transition_smooth _ _ (fun s => productLineCore_transition_smooth K (label s))))
/-- The finite-coordinate presentation is the same actual bundle, through
an explicit base-preserving linear homeomorphism of total spaces. -/
def finiteCoordinateTotalHomeomorph :
TotalSpace (EuclideanSpace ℂ S) (SmoothLineSumBundle K label) ≃ₜ
TotalSpace (Fin d → ℂ) (FiniteCoordinateBundle K label d hd) :=
changeModelCoreTotalHomeomorph _ _
end Kaplansky
end
open Bundle Topology
namespace Kaplansky
open ComplexGrassmann
variable {S ι : Type*} [Fintype S] [Fintype ι]
(K : ι → ℕ) (label : S → Option ι) (d : ℕ) (hd : Fintype.card S=d)
/-- Pull back the Hilbert norm, keeping the already constructed fiber topology. -/
instance (priority := 2000) finiteCoordinateHilbertNorm (P : ProjectiveProduct K) :
NormedAddCommGroup (FiniteCoordinateBundle K label d hd P) := by
let e := (finiteCoordinateEquiv d hd).symm
let N := NormedAddCommGroup.induced (FiniteCoordinateBundle K label d hd P)
(EuclideanSpace ℂ S) e.toLinearEquiv.toAddEquiv e.injective
exact { N.replaceTopology e.toHomeomorph.isInducing.eq_induced with
norm := N.norm
dist_eq := N.dist_eq }
instance (priority := 2000) finiteCoordinateHilbertInner (P : ProjectiveProduct K) :
InnerProductSpace ℂ (FiniteCoordinateBundle K label d hd P) := by
let e : FiniteCoordinateBundle K label d hd P ≃ₗ[ℂ] EuclideanSpace ℂ S :=
(finiteCoordinateEquiv d hd).symm.toLinearEquiv
exact
{ norm_smul_le := fun z v => by
change ‖e (z • v)‖ ≤ ‖z‖ * ‖e v‖
rw [map_smul, norm_smul]
inner := fun v w => inner ℂ (e v) (e w)
add_left := fun v w z => by simp only [map_add, inner_add_left]
smul_left := fun v w z => by simp only [map_smul, inner_smul_left]
norm_sq_eq_re_inner := fun v => norm_sq_eq_re_inner (e v)
conj_inner_symm := fun v w => inner_conj_symm _ _ }
omit [Fintype ι] in
/-- The actual sum metric is continuous in the finite-coordinate presentation. -/
theorem finiteCoordinate_continuousHermitianMetric :
ContinuousHermitianMetric (d := d) (E := FiniteCoordinateBundle K label d hd) := by
intro A v w hv hw
let H := finiteCoordinateTotalHomeomorph K label d hd
have hv' := H.symm.continuous.comp_continuousOn hv
have hw' := H.symm.continuous.comp_continuousOn hw
exact smoothLineSum_inner_continuousOn K label A
(fun P => (finiteCoordinateEquiv d hd).symm (v P))
(fun P => (finiteCoordinateEquiv d hd).symm (w P)) hv' hw'
end Kaplansky
end
end OAI
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