| import Mathlib |
|
|
| 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 |
| |
