| import OAI.Geometry.Relativity.CKS.ComparatorDefinitions |
| import OAI.Geometry.Relativity.CKS.SchwarzschildCharts |
| import OAI.Geometry.Relativity.CKS.LocalConstraints |
| import OAI.Geometry.Relativity.CKS.VolumeDefinitions |
| import OAI.Geometry.Relativity.CKS.BoundaryPartial |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Manifold Bundle Filter Function |
| open scoped ContDiff Topology |
| namespace CKSBoundarySurface |
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace H3 M] [IsManifold I3 ∞ M] |
|
|
| omit [IsManifold I3 ∞ M] in |
| lemma ext_symm_lift [IsManifold I3 ∞ M] (x : M) (z : E2) : |
| (extChartAt I3 x).symm (liftPlane z) = (chartAt H3 x).symm (liftHalf z) := by |
| change (chartAt H3 x).symm (I3.symm (I3 (liftHalf z))) = _ |
| rw [I3.left_inv] |
| omit [IsManifold I3 ∞ M] in |
| lemma lift_mem_ext_target [IsManifold I3 ∞ M] (x : M) {z : E2} |
| (hz : liftHalf z ∈ (chartAt H3 x).target) : |
| liftPlane z ∈ (extChartAt I3 x).target := by |
| change liftPlane z ∈ ((chartAt H3 x).extend I3).target |
| rw [OpenPartialHomeomorph.extend_target'] |
| exact ⟨liftHalf z,hz,rfl⟩ |
|
|
| lemma transition_lift_maps (x y : Boundary M) : |
| MapsTo liftPlane ((boundaryChart x).symm ≫ₕ boundaryChart y).source |
| ((extChartAt I3 x.val).symm ≫ extChartAt I3 y.val).source := by |
| intro z hz |
| change z ∈ (boundaryChart x).target ∧ (boundaryChart x).symm z ∈ (boundaryChart y).source at hz |
| refine ⟨lift_mem_ext_target x.val hz.1, ?_⟩ |
| change (extChartAt I3 x.val).symm (liftPlane z) ∈ (extChartAt I3 y.val).source |
| rw [ext_symm_lift] |
| have h := hz.2 |
| change ((boundaryChart x).symm z).val ∈ (chartAt H3 y.val).source at h |
| rw [boundaryChart_symm_val x hz.1] at h |
| simpa only [extChartAt_source] using h |
|
|
| lemma transition_formula (x y : Boundary M) {z : E2} |
| (hz : z ∈ ((boundaryChart x).symm ≫ₕ boundaryChart y).source) : |
| ((boundaryChart x).symm ≫ₕ boundaryChart y) z = |
| dropPlane (((extChartAt I3 x.val).symm ≫ extChartAt I3 y.val) (liftPlane z)) := by |
| change dropPlane (I3 (chartAt H3 y.val ((boundaryChart x).symm z).val)) = _ |
| rw [boundaryChart_symm_val x hz.1] |
| change _ = dropPlane (I3 (chartAt H3 y.val ((extChartAt I3 x.val).symm (liftPlane z)))) |
| rw [ext_symm_lift] |
|
|
| lemma inclusion_chart_formula (x : Boundary M) {z : E2} |
| (hz : z ∈ (boundaryChart x).target) : |
| writtenInExtChartAt I2 I3 x (Subtype.val : Boundary M → M) z = liftPlane z := by |
| change I3 (chartAt H3 x.val (((boundaryChart x).symm z).val)) = _ |
| rw [boundaryChart_symm_val x hz, (chartAt H3 x.val).right_inv hz] |
| rfl |
|
|
| lemma inclusion_chart_eventually (x : Boundary M) : |
| writtenInExtChartAt I2 I3 x (Subtype.val : Boundary M → M) =ᶠ[𝓝 (extChartAt I2 x x)] liftPlane := by |
| have ht : (boundaryChart x).target ∈ 𝓝 (extChartAt I2 x x) := |
| (boundaryChart x).open_target.mem_nhds ((boundaryChart x).map_source (mem_chart_source E2 x)) |
| filter_upwards [ht] with z hz |
| exact inclusion_chart_formula x hz |
|
|
| lemma inclusion_smooth : ContMDiff I2 I3 ∞ (Subtype.val : Boundary M → M) := by |
| intro x |
| apply contMDiffAt_iff.mpr |
| refine ⟨continuous_subtype_val.continuousAt, ?_⟩ |
| exact ((liftPlane.contDiff.contDiffAt).congr_of_eventuallyEq (inclusion_chart_eventually x)).contDiffWithinAt |
|
|
| lemma inclusion_hasMFDeriv (x : Boundary M) : |
| HasMFDerivAt I2 I3 (Subtype.val : Boundary M → M) x liftPlane := by |
| refine ⟨continuous_subtype_val.continuousAt, ?_⟩ |
| exact (liftPlane.hasFDerivAt.congr_of_eventuallyEq (inclusion_chart_eventually x)).hasFDerivWithinAt |
|
|
| lemma inclusion_mfderiv_injective (x : Boundary M) : |
| Injective (mfderiv I2 I3 (Subtype.val : Boundary M → M) x) := by |
| rw [(inclusion_hasMFDeriv x).mfderiv] |
| intro a b h |
| change liftPlane a = liftPlane b at h |
| have hh : dropPlane (liftPlane a) = dropPlane (liftPlane b) := congrArg dropPlane h |
| change (a : E2) = (b : E2) |
| exact (drop_lift a).symm.trans (hh.trans (drop_lift b)) |
|
|
| end CKSBoundarySurface |
|
|
| end |
|
|
| end OAI |
|
|
