| import OAI.Geometry.Immersion.ClosedSurface.MetricPatches | |
| import OAI.Geometry.Immersion.ClosedSurface.JacobianBounds | |
| namespace OAI | |
| noncomputable section | |
| open Set Complex Bundle Manifold | |
| open scoped ContDiff Matrix Topology Manifold BigOperators | |
| namespace ClosedSurfaceR4 | |
| open SmallModes RealModes PhaseGeometry Set Filter | |
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] | |
| def coordinateChart (p : M) : OpenPartialHomeomorph M SmallModes.Base := | |
| (chartAt Plane p).trans planeCoordinates.toHomeomorph.toOpenPartialHomeomorph | |
| lemma coordinateChart_source (p : M) : | |
| (coordinateChart p).source = (chartAt Plane p).source := by | |
| simp [coordinateChart] | |
| lemma coordinateChart_apply (p q : M) : coordinateChart p q = chartCoordinates p q := rfl | |
| lemma coordinateChart_symm_apply (p : M) (x : SmallModes.Base) : | |
| (coordinateChart p).symm x = coordinateInverse p x := by | |
| simp [coordinateChart,coordinateInverse,planeModel] | |
| lemma coordinateChart_target (p : M) : | |
| (coordinateChart p).target = coordinateDomain p := by | |
| simp [coordinateChart,coordinateDomain,planeModel] | |
| variable [IsManifold planeModel ∞ M] | |
| lemma coordinateChart_smoothOn (p : M) : | |
| ContMDiffOn planeModel 𝓘(ℝ,SmallModes.Base) ∞ (coordinateChart p) (coordinateChart p).source := by | |
| rw [coordinateChart_source] | |
| exact planeCoordinates.contDiff.contMDiff.comp_contMDiffOn contMDiffOn_chart | |
| lemma coordinateChart_symm_smoothOn (p : M) : | |
| ContMDiffOn 𝓘(ℝ,SmallModes.Base) planeModel ∞ (coordinateChart p).symm (coordinateChart p).target := by | |
| have he : ((coordinateChart p).symm : SmallModes.Base → M) = coordinateInverse p := by | |
| funext x; exact coordinateChart_symm_apply p x | |
| rw [he,coordinateChart_target] | |
| exact coordinateInverse_smoothOn p | |
| def coordinateTransition (q p : M) : OpenPartialHomeomorph SmallModes.Base SmallModes.Base := | |
| (coordinateChart p).symm.trans (coordinateChart q) | |
| lemma coordinateTransition_smoothOn (q p : M) : | |
| ContDiffOn ℝ ∞ (coordinateTransition q p) (coordinateTransition q p).source := by | |
| apply ContMDiffOn.contDiffOn | |
| exact (coordinateChart_smoothOn q).comp | |
| ((coordinateChart_symm_smoothOn p).mono (fun x hx => hx.1)) (fun x hx => hx.2) | |
| omit [IsManifold planeModel ∞ M] in | |
| lemma coordinateTransition_symm (q p : M) : | |
| (coordinateTransition q p).symm = coordinateTransition p q := by | |
| simp [coordinateTransition,OpenPartialHomeomorph.trans_symm_eq_symm_trans_symm] | |
| lemma coordinateTransition_symm_smoothOn (q p : M) : | |
| ContDiffOn ℝ ∞ (coordinateTransition q p).symm (coordinateTransition q p).target := by | |
| have h := coordinateTransition_smoothOn p q | |
| rw [← coordinateTransition_symm q p] at h | |
| exact h | |
| theorem coordinateTransition_compact_bounds (q p : M) {K : Set SmallModes.Base} | |
| (hK : IsCompact K) (hKs : K ⊆ (coordinateTransition q p).source) : | |
| ∃ d C : ℝ, 0 < d ∧ 0 < C ∧ | |
| ∀ x ∈ K, d ≤ |coordDet (fderiv ℝ (coordinateTransition q p) x)| ∧ | |
| ‖fderiv ℝ (coordinateTransition q p) x‖ ≤ C := by | |
| let e := coordinateTransition q p | |
| exact compact_chart_jacobian_bounds e.open_source e.open_target | |
| (coordinateTransition_smoothOn q p) (coordinateTransition_symm_smoothOn q p) | |
| e.mapsTo (fun x hx => e.left_inv hx) hK hKs | |
| end ClosedSurfaceR4 | |
| end | |
| end OAI | |