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namespace OAI
/-!
# Smooth metrics, Jacobi fields, and the three-manifold existence statement.
-/
noncomputable section
open Set Manifold Bundle
open scoped ContDiff
namespace ThreeManifold
abbrev E3 := Fin 3 → ℝ
abbrev I3 := 𝓘(ℝ, E3)
variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M]
def OrientedAtlas : Prop :=
∀ (c d : OpenPartialHomeomorph M E3), c ∈ atlas E3 M → d ∈ atlas E3 M →
∀ x ∈ c.target, c.symm x ∈ d.source →
0 < LinearMap.det (fderiv ℝ (d ∘ c.symm) x).toLinearMap
variable [IsManifold I3 ∞ M]
abbrev SmoothMetric (M : Type) [TopologicalSpace M] [ChartedSpace E3 M]
[IsManifold I3 ∞ M] :=
ContMDiffRiemannianMetric I3 ∞ E3 (fun x : M => TangentSpace I3 x)
def basisVector (i : Fin 3) : E3 := Pi.single i 1
def metricMatrix (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3) (x : E3) :
Matrix (Fin 3) (Fin 3) ℝ :=
fun i j => g.inner (c.symm x)
(mfderiv I3 I3 c.symm x (basisVector i))
(mfderiv I3 I3 c.symm x (basisVector j))
def coordPartial (f : E3 → ℝ) (x : E3) (i : Fin 3) : ℝ :=
fderiv ℝ f x (basisVector i)
def christoffel (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)
(x : E3) (i j k : Fin 3) : ℝ :=
(1 / 2 : ℝ) * ∑ l : Fin 3, (metricMatrix g c x)⁻¹ i l *
(coordPartial (fun y => metricMatrix g c y l k) x j +
coordPartial (fun y => metricMatrix g c y l j) x k -
coordPartial (fun y => metricMatrix g c y j k) x l)
def connectionTerm (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)
(x u v : E3) : E3 :=
fun i => ∑ j : Fin 3, ∑ k : Fin 3, christoffel g c x i j k * u j * v k
def curvatureCoefficient (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)
(x : E3) (l k i j : Fin 3) : ℝ :=
coordPartial (fun y => christoffel g c y l j k) x i -
coordPartial (fun y => christoffel g c y l i k) x j +
∑ a : Fin 3, (christoffel g c x l i a * christoffel g c x a j k -
christoffel g c x l j a * christoffel g c x a i k)
def curvatureTerm (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)
(x u v w : E3) : E3 :=
fun l => ∑ k : Fin 3, ∑ i : Fin 3, ∑ j : Fin 3,
curvatureCoefficient g c x l k i j * w k * u i * v j
def NonpositiveSectionalCurvature (g : SmoothMetric M) : Prop :=
∀ (c : OpenPartialHomeomorph M E3), c ∈ atlas E3 M →
∀ x ∈ c.target, ∀ u v : E3,
(∑ i : Fin 3, ∑ j : Fin 3,
metricMatrix g c x i j * curvatureTerm g c x u v v i * u j) ≤ 0
def alongDerivative (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)
(x w : ℝ → E3) (s : ℝ) : E3 :=
deriv w s + connectionTerm g c (x s) (deriv x s) (w s)
def IsGeodesicOn (g : SmoothMetric M) (γ : ℝ → M) (U : Set ℝ) : Prop :=
ContMDiffOn 𝓘(ℝ, ℝ) I3 ∞ γ U ∧
∀ (c : OpenPartialHomeomorph M E3), c ∈ atlas E3 M →
∀ s ∈ U, γ s ∈ c.source →
alongDerivative g c (c ∘ γ) (deriv (c ∘ γ)) s = 0
def fieldCoordinates (γ : ℝ → M) (J : ∀ s, TangentSpace I3 (γ s))
(c : OpenPartialHomeomorph M E3) (s : ℝ) : E3 :=
mfderiv I3 I3 c (γ s) (J s)
def IsJacobiFieldOn (g : SmoothMetric M) (γ : ℝ → M)
(J : ∀ s, TangentSpace I3 (γ s)) (U : Set ℝ) : Prop :=
∀ (c : OpenPartialHomeomorph M E3), c ∈ atlas E3 M →
ContDiffOn ℝ ∞ (fieldCoordinates γ J c) (U ∩ γ ⁻¹' c.source) ∧
∀ s ∈ U, γ s ∈ c.source →
alongDerivative g c (c ∘ γ)
(alongDerivative g c (c ∘ γ) (fieldCoordinates γ J c)) s +
curvatureTerm g c (c (γ s)) (fieldCoordinates γ J c s)
(deriv (c ∘ γ) s) (deriv (c ∘ γ) s) = 0
def NoConjugatePoints (g : SmoothMetric M) : Prop :=
∀ (U : Set ℝ), IsOpen U → OrdConnected U →
∀ (γ : ℝ → M), IsGeodesicOn g γ U →
∀ (J : ∀ s, TangentSpace I3 (γ s)), IsJacobiFieldOn g γ J U →
∀ a ∈ U, ∀ b ∈ U, a ≠ b → J a = 0 → J b = 0 → ∀ s ∈ U, J s = 0
def MainStatement : Prop :=
∃ (M : Type) (top : TopologicalSpace M),
letI := top
∃ (_ : T2Space M) (_ : SecondCountableTopology M) (_ : CompactSpace M)
(_ : ConnectedSpace M) (charts : ChartedSpace E3 M),
letI := charts
∃ (smooth : IsManifold I3 ∞ M),
letI := smooth
OrientedAtlas (M := M) ∧
(∃ g : SmoothMetric M, NoConjugatePoints g) ∧
¬ ∃ g : SmoothMetric M, NonpositiveSectionalCurvature g
end ThreeManifold
end
end OAI
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