| import OAI.Geometry.Immersion.ClosedSurface.RealModes |
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| namespace OAI |
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| noncomputable section |
| open Set Complex Bundle Manifold |
| open scoped ContDiff Matrix Topology Manifold BigOperators |
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| namespace ClosedSurfaceR4 |
| open SmallModes RealModes Set Bundle Manifold |
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| def planeCoordinates : Plane ≃L[ℝ] Base := |
| (EuclideanSpace.equiv (Fin 2) ℝ).trans |
| (ContinuousLinearEquiv.piFinTwo ℝ (fun _ => ℝ)) |
| def spaceCoordinates : Space ≃L[ℝ] RVec 4 := EuclideanSpace.equiv (Fin 4) ℝ |
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| variable {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] |
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| def coordinateDomain (p : M) : Set Base := |
| planeCoordinates.symm ⁻¹' (extChartAt planeModel p).target |
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| def coordinateCenter (p : M) : Base := planeCoordinates (extChartAt planeModel p p) |
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| def coordinateMap (F : M → Space) (p : M) : RField 4 := |
| spaceCoordinates ∘ F ∘ (extChartAt planeModel p).symm ∘ planeCoordinates.symm |
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| lemma coordinateDomain_open (p : M) : IsOpen (coordinateDomain p) := by |
| apply IsOpen.preimage planeCoordinates.symm.continuous |
| simpa [extChartAt_target, planeModel] using (chartAt Plane p).open_target |
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| lemma coordinateCenter_mem (p : M) : coordinateCenter p ∈ coordinateDomain p := by |
| simp [coordinateCenter, coordinateDomain] |
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| lemma coordinateMap_at_center (F : M → Space) (p : M) : |
| coordinateMap F p (coordinateCenter p) = spaceCoordinates (F p) := by |
| simp [coordinateMap, coordinateCenter] |
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| lemma coordinateMap_smoothOn [IsManifold planeModel ∞ M] {F : M → Space} |
| (hF : ContMDiff planeModel spaceModel ∞ F) (p : M) : |
| ContDiffOn ℝ ∞ (coordinateMap F p) (coordinateDomain p) := by |
| apply ContMDiffOn.contDiffOn |
| have hChart := contMDiffOn_extChartAt_symm (I := planeModel) (n := (∞ : ℕ∞ω)) p |
| have h1 := hF.comp_contMDiffOn hChart |
| have h2 := spaceCoordinates.contDiff.contMDiff.comp_contMDiffOn h1 |
| exact h2.comp planeCoordinates.symm.contDiff.contMDiff.contMDiffOn |
| (fun x hx => hx) |
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| lemma mfderiv_coordinateMap_center {F : M → Space} |
| (hF : ContMDiff planeModel spaceModel ∞ F) (p : M) : |
| fderiv ℝ (coordinateMap F p) (coordinateCenter p) = |
| spaceCoordinates.toContinuousLinearMap.comp |
| ((mfderiv planeModel spaceModel F p).comp |
| planeCoordinates.symm.toContinuousLinearMap) := by |
| have hd := (hF p).mdifferentiableAt (by simp) |
| have heq : mfderiv planeModel spaceModel F p = |
| fderiv ℝ (F ∘ (extChartAt planeModel p).symm) (extChartAt planeModel p p) := by |
| simpa [writtenInExtChartAt, planeModel, spaceModel, chartAt_self_eq, |
| differentiableWithinAt_univ] using hd.mfderiv_abuse |
| have h0 : DifferentiableAt ℝ (F ∘ (extChartAt planeModel p).symm) |
| (extChartAt planeModel p p) := by |
| simpa [writtenInExtChartAt, planeModel, spaceModel, chartAt_self_eq, |
| differentiableWithinAt_univ] using |
| hd.differentiableWithinAt_writtenInExtChartAt |
| have h1 : HasFDerivAt (spaceCoordinates ∘ F ∘ (extChartAt planeModel p).symm) |
| (spaceCoordinates.toContinuousLinearMap.comp |
| (fderiv ℝ (F ∘ (extChartAt planeModel p).symm) (extChartAt planeModel p p))) |
| (planeCoordinates.symm (coordinateCenter p)) := by |
| simpa [coordinateCenter] using spaceCoordinates.hasFDerivAt.comp |
| (extChartAt planeModel p p) h0.hasFDerivAt |
| have hc := h1.comp (coordinateCenter p) planeCoordinates.symm.hasFDerivAt |
| rw [heq] |
| have hc' := hc.fderiv |
| simp only [coordinateMap, Function.comp_assoc, ContinuousLinearMap.comp_assoc] at hc' ⊢ |
| exact hc' |
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| lemma coordinateMap_immersion_center {F : M → Space} |
| (hF : ContMDiff planeModel spaceModel ∞ F) (p : M) |
| (hImm : Function.Injective (mfderiv planeModel spaceModel F p)) : |
| Function.Injective (fderiv ℝ (coordinateMap F p) (coordinateCenter p)) := by |
| rw [mfderiv_coordinateMap_center hF p] |
| exact spaceCoordinates.injective.comp (hImm.comp planeCoordinates.symm.injective) |
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| lemma spaceCoordinates_dot (v w : Space) : |
| spaceCoordinates v ⬝ᵥ spaceCoordinates w = inner ℝ v w := by |
| simp [spaceCoordinates, dotProduct, PiLp.inner_apply, mul_comm] |
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| lemma coordinateMetric_at_center {F : M → Space} |
| (hF : ContMDiff planeModel spaceModel ∞ F) (p : M) (v w : Base) : |
| (fderiv ℝ (coordinateMap F p) (coordinateCenter p) v) ⬝ᵥ |
| (fderiv ℝ (coordinateMap F p) (coordinateCenter p) w) = |
| inner ℝ (mfderiv planeModel spaceModel F p (planeCoordinates.symm v)) |
| (mfderiv planeModel spaceModel F p (planeCoordinates.symm w)) := by |
| rw [mfderiv_coordinateMap_center hF p] |
| exact spaceCoordinates_dot _ _ |
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| end ClosedSurfaceR4 |
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| end |
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| end OAI |
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