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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | import OAI.Geometry.Immersion.ClosedSurface.RealModes
namespace OAI
noncomputable section
open Set Complex Bundle Manifold
open scoped ContDiff Matrix Topology Manifold BigOperators
namespace ClosedSurfaceR4
open SmallModes RealModes Set Bundle Manifold
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
variable {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M]
def coordinateDomain (p : M) : Set Base :=
planeCoordinates.symm ⁻¹' (extChartAt planeModel p).target
def coordinateCenter (p : M) : Base := planeCoordinates (extChartAt planeModel p p)
def coordinateMap (F : M → Space) (p : M) : RField 4 :=
spaceCoordinates ∘ F ∘ (extChartAt planeModel p).symm ∘ planeCoordinates.symm
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
lemma coordinateCenter_mem (p : M) : coordinateCenter p ∈ coordinateDomain p := by
simp [coordinateCenter, coordinateDomain]
lemma coordinateMap_at_center (F : M → Space) (p : M) :
coordinateMap F p (coordinateCenter p) = spaceCoordinates (F p) := by
simp [coordinateMap, coordinateCenter]
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)
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'
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)
lemma spaceCoordinates_dot (v w : Space) :
spaceCoordinates v ⬝ᵥ spaceCoordinates w = inner ℝ v w := by
simp [spaceCoordinates, dotProduct, PiLp.inner_apply, mul_comm]
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 _ _
end ClosedSurfaceR4
end
end OAI
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