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785c727 | 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 | import OAI.Geometry.EinsteinFour.MetricDoubleCover
import OAI.Geometry.EinsteinFour.FrameOrientation
import Mathlib.Geometry.Manifold.Category.MfldCat.OfModel
namespace OAI
/-! The topological reduction from the oriented two-model theorem to the
unoriented three-model theorem uses the orientation cover, its pullback metric
and the Euler/Betti identities from Hatcher. -/
namespace PositiveEinsteinFour
open EinsteinFour
open scoped Manifold ContDiff
noncomputable section
universe u
/-- A closed connected oriented Riemannian four-manifold, with its genuine
smooth manifold structure. The measurable structure can be installed as
the Borel structure when applying the analytic theorem. -/
structure ClosedOrientedMetricFour where
manifold : ModelWithCorners.MfldCat.{u} I4 ∞
[regular : T3Space manifold]
[secondCountable : SecondCountableTopology manifold]
[compact : CompactSpace manifold]
[connected : ConnectedSpace manifold]
metric : SmoothMetric manifold
orientation : Geometry.FrameOrientation metric
attribute [instance] ClosedOrientedMetricFour.regular
ClosedOrientedMetricFour.secondCountable ClosedOrientedMetricFour.compact
ClosedOrientedMetricFour.connected
variable {M : Type u} [TopologicalSpace M] [ChartedSpace FourSpace M]
[IsManifold I4 ∞ M]
/-- Hatcher's connected orientation cover, equipped with the pullback
metric. Preservation of Einstein and positive sectional curvature is the
local-isometry specialization in Petersen §1.3.3. -/
structure OrientationMetricDoubleCover (g : SmoothMetric M) where
oriented : ClosedOrientedMetricFour.{u}
cover : MetricDoubleCover g oriented.metric
einstein : IsEinstein g → IsEinstein oriented.metric
positive : HasPositiveSectionalCurvature g → HasPositiveSectionalCurvature oriented.metric
/-- Hatcher, Proposition 3.25 and Theorem 3.26(c), together with smooth
metric pullback: either an orientation exists, or its connected two-sheeted
cover has the displayed actual geometric data. -/
def HatcherOrientationCoverInput (g : SmoothMetric M) : Prop :=
Nonempty (Geometry.FrameOrientation g) ∨ Nonempty (OrientationMetricDoubleCover g)
/-- The precise covering/topological inputs needed after the oriented
classification. The Euler and Betti statements concern the actual cover
manifold; the antipodal quotient is proved from its metric data. -/
structure UnorientedPublishedInputs (g : SmoothMetric M) : Prop where
orientationCover : HatcherOrientationCoverInput g
euler : ∀ C : OrientationMetricDoubleCover g,
HatcherDoubleEulerInput (M := M) (N := C.oriented.manifold)
projectiveBetti : ∀ C : OrientationMetricDoubleCover g,
HatcherProjectiveBettiInput C.oriented.metric
variable [T3Space M] [SecondCountableTopology M] [CompactSpace M] [ConnectedSpace M]
/-- The oriented analytic classification, once supplied for every closed
oriented metric, implies the full unoriented classification. The sphere
cover descends through a proved free-involution quotient; the CP² cover
contradicts the proved odd Euler obstruction. -/
theorem classified_of_oriented_models (g : SmoothMetric M)
(hpublished : UnorientedPublishedInputs g)
(hmodels : ∀ X : ClosedOrientedMetricFour.{u},
IsEinstein X.metric → HasPositiveSectionalCurvature X.metric →
∃ a : ℝ, 0 < a ∧ (ScaledIsometricTo X.metric a roundDistance ∨
ScaledIsometricTo X.metric a fubiniStudyDistance))
(hEinstein : IsEinstein g) (hpositive : HasPositiveSectionalCurvature g) :
Classified g := by
rcases hpublished.orientationCover with ho | hC
· obtain ⟨o⟩ := ho
let X : ClosedOrientedMetricFour.{u} :=
{ manifold := ModelWithCorners.MfldCat.of M
metric := g
orientation := o }
obtain ⟨a, ha, hs | hp⟩ := hmodels X hEinstein hpositive
· exact ⟨a, ha, Or.inl hs⟩
· exact ⟨a, ha, Or.inr (Or.inl hp)⟩
· obtain ⟨C⟩ := hC
exact C.cover.classified_of_two_models (hpublished.euler C)
(hpublished.projectiveBetti C)
(hmodels C.oriented (C.einstein hEinstein) (C.positive hpositive))
end
end PositiveEinsteinFour
end OAI
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