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openAI-math-dataset / lean /OAI /Geometry /ComplexBundle /Coordinates.lean
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Add lean source and binary files (part 66)
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import OAI.Analysis.MatrixScaling.Expansion
import OAI.Geometry.ComplexBundle.Hom
namespace OAI
noncomputable section
/-! Coordinates and realization of Hermitian scaling sections. -/
section
open scoped BigOperators ComplexOrder MatrixOrder Matrix.Norms.L2Operator
open Matrix Bundle Topology
namespace Kaplansky
variable {n d r : ℕ} {Y : Type*} [TopologicalSpace Y]
variable {E F : Y → Type*}
variable [∀ y, NormedAddCommGroup (E y)] [∀ y, InnerProductSpace ℂ (E y)]
variable [∀ y, NormedAddCommGroup (F y)] [∀ y, InnerProductSpace ℂ (F y)]
variable [TopologicalSpace (TotalSpace (Fin d → ℂ) E)]
variable [TopologicalSpace (TotalSpace (Fin r → ℂ) F)]
variable [FiberBundle (Fin d → ℂ) E] [VectorBundle ℂ (Fin d → ℂ) E]
variable [FiberBundle (Fin r → ℂ) F] [VectorBundle ℂ (Fin r → ℂ) F]
/-- Coordinates of genuine fiber maps in the constructed orthonormal frames. -/
def bundleMapCoordinates (T : ∀ y, Fin n → E y →ₗ[ℂ] F y) (a y : Y) :
Fin n → Matrix (Fin r) (Fin d) ℂ :=
fun i k j => inner ℂ
(bundleOrthonormalFrame (trivializationAt (Fin r → ℂ) F a) y k)
(T y i (bundleOrthonormalFrame (trivializationAt (Fin d → ℂ) E a) y j))
theorem bundleMapCoordinates_eq (T : ∀ y, Fin n → E y →ₗ[ℂ] F y) (a y : Y)
(hy : y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) a) (i : Fin n) :
bundleMapCoordinates (d := d) (r := r) T a y i = LinearMap.toMatrix
(bundleOrthonormalBasis (trivializationAt (Fin d → ℂ) E a) y hy.1).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin r → ℂ) F a) y hy.2).toBasis (T y i) := by
ext k j
simp only [bundleMapCoordinates, LinearMap.toMatrix_apply,
OrthonormalBasis.coe_toBasis_repr_apply, OrthonormalBasis.repr_apply_apply,
OrthonormalBasis.coe_toBasis, bundleOrthonormalBasis_apply]
theorem continuousOn_bundleMapCoordinates
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(T : ∀ y, Fin n → E y →ₗ[ℂ] F y)
(hT : ∀ i, Continuous (fun z : TotalSpace (Fin d → ℂ) E =>
TotalSpace.mk' (Fin r → ℂ) z.proj (T z.proj i z.2))) (a : Y) :
ContinuousOn (bundleMapCoordinates (d := d) (r := r) T a) (hermitianChartSet (d := d) (r := r) (E := E) (F := F) a) := by
apply continuousOn_pi.mpr
intro i
apply continuousOn_pi.mpr
intro k
apply continuousOn_pi.mpr
intro j
apply hF
· exact (continuousOn_bundleOrthonormalFrame hF _ k).mono Set.inter_subset_right
· exact (hT i).comp_continuousOn
((continuousOn_bundleOrthonormalFrame hE _ j).mono Set.inter_subset_left)
theorem bundleMapCoordinates_expanding
(T : ∀ y, Fin n → E y →ₗ[ℂ] F y) (hT : ∀ y, IntrinsicExpanding (T y))
(a y : Y) (hy : y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) a) :
Expanding (bundleMapCoordinates (d := d) (r := r) T a y) := by
have he : bundleMapCoordinates (d := d) (r := r) T a y = fun i => LinearMap.toMatrix
(bundleOrthonormalBasis (trivializationAt (Fin d → ℂ) E a) y hy.1).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin r → ℂ) F a) y hy.2).toBasis (T y i) :=
funext (bundleMapCoordinates_eq T a y hy)
rw [he]
exact expanding_in_basis _ _ _ (hT y)
theorem bundleUnitaryTransition_star
{k : ℕ} {V : Y → Type*} [∀ y, NormedAddCommGroup (V y)] [∀ y, InnerProductSpace ℂ (V y)]
[TopologicalSpace (TotalSpace (Fin k → ℂ) V)]
[FiberBundle (Fin k → ℂ) V] [VectorBundle ℂ (Fin k → ℂ) V]
(a b : Trivialization (Fin k → ℂ) (π (Fin k → ℂ) V)) [a.IsLinear ℂ] [b.IsLinear ℂ]
(y : Y) : (bundleUnitaryTransition a b y)ᴴ = bundleUnitaryTransition b a y := by
apply Matrix.ext
intro i j
exact inner_conj_symm _ _
theorem bundleMapCoordinates_compatible
(T : ∀ y, Fin n → E y →ₗ[ℂ] F y) (a b y : Y)
(ha : y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) a)
(hb : y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) b) :
bundleMapCoordinates (d := d) (r := r) T b y = frameTuple (bundleMapCoordinates (d := d) (r := r) T a y)
(hermitianSourceTransition (d := d) (E := E) a b y)
(hermitianTargetTransition (r := r) (F := F) a b y) := by
funext i
dsimp only [frameTuple, hermitianSourceTransition, hermitianTargetTransition]
rw [bundleUnitaryTransition_star, bundleMapCoordinates_eq T a y ha i,
bundleMapCoordinates_eq T b y hb i,
bundleUnitaryTransition_eq _ _ y hb.1 ha.1,
bundleUnitaryTransition_eq _ _ y ha.2 hb.2]
symm
exact basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix
(bundleOrthonormalBasis (trivializationAt (Fin d → ℂ) E b) y hb.1).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin d → ℂ) E a) y ha.1).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin r → ℂ) F b) y hb.2).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin r → ℂ) F a) y ha.2).toBasis (T y i)
omit [∀ y, InnerProductSpace ℂ (E y)] [∀ y, InnerProductSpace ℂ (F y)]
[VectorBundle ℂ (Fin d → ℂ) E] [VectorBundle ℂ (Fin r → ℂ) F] in
theorem mem_hermitianChartSet (y : Y) :
y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) y :=
⟨mem_baseSet_trivializationAt (Fin d → ℂ) E y,
mem_baseSet_trivializationAt (Fin r → ℂ) F y⟩
/-- Interpret a section of the constructed core as linear maps between
its original bundles, using the core's selected chart in each fiber. -/
def realizeHomSection
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(s : ∀ y, (hermitianHomCore (n := n) hE hF).Fiber y) (y : Y) (i : Fin n) :
E y →ₗ[ℂ] F y :=
Matrix.toLin
(bundleOrthonormalBasis (trivializationAt (Fin d → ℂ) E y) y
(mem_baseSet_trivializationAt (Fin d → ℂ) E y)).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin r → ℂ) F y) y
(mem_baseSet_trivializationAt (Fin r → ℂ) F y)).toBasis (s y i)
theorem realizeHomSection_at
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(s : ∀ y, (hermitianHomCore (n := n) hE hF).Fiber y) (y : Y) :
bundleMapCoordinates (d := d) (r := r) (realizeHomSection hE hF s) y y = s y := by
funext i
rw [bundleMapCoordinates_eq _ y y (mem_hermitianChartSet (d := d) (r := r) y) i]
exact LinearMap.toMatrix_toLin _ _ (s y i)
theorem realizeHomSection_coordinates
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(s : ∀ y, (hermitianHomCore (n := n) hE hF).Fiber y) (a y : Y)
(hy : y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) a) :
bundleMapCoordinates (d := d) (r := r) (realizeHomSection hE hF s) a y =
((hermitianHomCore hE hF).localTriv a ⟨y,s y⟩).2 := by
rw [bundleMapCoordinates_compatible _ y a y
(mem_hermitianChartSet (d := d) (r := r) y) hy, realizeHomSection_at]
rfl
theorem continuousOn_realizeHomSection_coordinates
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(s : ∀ y, (hermitianHomCore (n := n) hE hF).Fiber y)
(hs : Continuous (fun y => TotalSpace.mk' (Fin n → Matrix (Fin r) (Fin d) ℂ) y (s y)))
(a : Y) :
ContinuousOn (bundleMapCoordinates (d := d) (r := r) (realizeHomSection hE hF s) a)
(hermitianChartSet (d := d) (r := r) (E := E) (F := F) a) := by
have hc := ((hermitianHomCore hE hF).localTriv a).continuousOn.comp hs.continuousOn
(show Set.MapsTo (fun y => TotalSpace.mk' (Fin n → Matrix (Fin r) (Fin d) ℂ) y (s y))
(hermitianChartSet (d := d) (r := r) (E := E) (F := F) a)
((hermitianHomCore hE hF).localTriv a).source from fun y hy =>
((hermitianHomCore hE hF).localTriv a).mem_source.mpr hy)
exact hc.snd.congr (fun y hy => realizeHomSection_coordinates hE hF s a y hy)
end Kaplansky
end
open scoped BigOperators ComplexOrder MatrixOrder Matrix.Norms.L2Operator
open Matrix Bundle Topology
namespace Kaplansky
variable {n d r : ℕ} {Y : Type*} [TopologicalSpace Y] [LocallyCompactSpace Y]
variable {E F : Y → Type*}
variable [∀ y, NormedAddCommGroup (E y)] [∀ y, InnerProductSpace ℂ (E y)]
variable [∀ y, NormedAddCommGroup (F y)] [∀ y, InnerProductSpace ℂ (F y)]
variable [TopologicalSpace (TotalSpace (Fin d → ℂ) E)]
variable [TopologicalSpace (TotalSpace (Fin r → ℂ) F)]
variable [FiberBundle (Fin d → ℂ) E] [VectorBundle ℂ (Fin d → ℂ) E]
variable [FiberBundle (Fin r → ℂ) F] [VectorBundle ℂ (Fin r → ℂ) F]
/-- Normalization of intrinsic expanding maps descends in the constructed
Hom-tuple bundle. The ordinary input bundles may be nontrivial. -/
theorem exists_hermitian_scaled_section
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(T : ∀ y, Fin n → E y →ₗ[ℂ] F y)
(hT : ∀ i, Continuous (fun z : TotalSpace (Fin d → ℂ) E =>
TotalSpace.mk' (Fin r → ℂ) z.proj (T z.proj i z.2)))
(hexp : ∀ y, IntrinsicExpanding (T y)) :
∃ s : ∀ y, (hermitianHomCore (n := n) hE hF).Fiber y,
Continuous (fun y => TotalSpace.mk' (Fin n → Matrix (Fin r) (Fin d) ℂ) y (s y)) ∧
∀ a y, y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) a →
let X := ((hermitianHomCore hE hF).localTriv a ⟨y,s y⟩).2
(∑ i, (X i)ᴴ * X i) = 1 ∧
((2/3 : ℂ) • (1 : Matrix (Fin r) (Fin r) ℂ) - ∑ i, X i * (X i)ᴴ).PosSemidef := by
exact exists_scaled_core_section (hermitianHomCore hE hF)
(bundleMapCoordinates (d := d) (r := r) T)
(continuousOn_bundleMapCoordinates hE hF T hT)
(bundleMapCoordinates_expanding T hexp)
(hermitianSourceTransition (d := d) (E := E))
(hermitianTargetTransition (r := r) (F := F))
(hermitianHomCore_unitary (d := d) (r := r))
(fun a b y _ _ M => hermitianHomCore_coordChange hE hF a b y M)
(bundleMapCoordinates_compatible T)
end Kaplansky
end
end OAI