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import OAI.Geometry.BallPacking.Forms.ManifoldMapDifferential
namespace OAI
noncomputable section
namespace PackingSufficiencySupport.Hamiltonian
open Set Function Manifold
open scoped ContDiff Manifold Topology
section
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
{M N : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M]
[TopologicalSpace N] [ChartedSpace F N] [IsManifold 𝓘(ℝ,F) ∞ N]
omit [IsManifold 𝓘(ℝ,E) ∞ M] in
theorem chartOneForm_manifoldPullback_at {α : ℝ → ManifoldOneForm E M} {e : N → M}
{b : N} {p : ℝ × F}
(hp : p.2 ∈ (extChartAt 𝓘(ℝ,F) b).target)
(he : MDifferentiableAt 𝓘(ℝ,F) 𝓘(ℝ,E) e ((extChartAt 𝓘(ℝ,F) b).symm p.2)) :
chartOneForm (manifoldPullbackOneForm α e p.1) b p.2 =
euclideanPullbackOneForm α (e ∘ (extChartAt 𝓘(ℝ,F) b).symm) p := by
have hc := ((contMDiffOn_extChartAt_symm (I := 𝓘(ℝ,F)) (n := ∞) b).contMDiffAt
((isOpen_extChartAt_target (I := 𝓘(ℝ,F)) b).mem_nhds hp)).mdifferentiableAt (by simp)
have hd := (he.hasMFDerivAt.comp p.2 hc.hasMFDerivAt).mfderiv
simp only [euclideanPullbackOneForm,hd,chartOneForm,manifoldPullbackOneForm,
manifoldMapDifferential,chartDifferential_inverse hp,Function.comp_apply]
ext v
rfl
theorem manifoldPullbackOneForm_smooth_on {α : ℝ → ManifoldOneForm E M}
{e : N → M} {U : Set N} (hU : IsOpen U)
(he : ContMDiffOn 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ e U)
(hα : ∀ c, ContDiffOn ℝ ∞ (fun q : ℝ × E => chartOneForm (α q.1) c q.2)
(univ ×ˢ (extChartAt 𝓘(ℝ,E) c).target)) (b : N) :
ContDiffOn ℝ ∞ (fun q : ℝ × F => chartOneForm (manifoldPullbackOneForm α e q.1) b q.2)
(univ ×ˢ ((extChartAt 𝓘(ℝ,F) b).target ∩ (extChartAt 𝓘(ℝ,F) b).symm ⁻¹' U)) := by
intro p hp
have hi := (contMDiffOn_extChartAt_symm (I := 𝓘(ℝ,F)) (n := ∞) b).contMDiffAt
((isOpen_extChartAt_target (I := 𝓘(ℝ,F)) b).mem_nhds hp.2.1)
have hg := (he.contMDiffAt (hU.mem_nhds hp.2.2)).comp p.2 hi
apply ((euclideanPullbackOneForm_contDiffAt hα hg).congr_of_eventuallyEq _).contDiffWithinAt
have ht := continuousAt_snd.preimage_mem_nhds
((isOpen_extChartAt_target (I := 𝓘(ℝ,F)) b).mem_nhds hp.2.1)
have hu := (hi.continuousAt.comp continuousAt_snd).preimage_mem_nhds (hU.mem_nhds hp.2.2)
filter_upwards [ht,hu] with q hqt hqu
exact chartOneForm_manifoldPullback_at hqt
((he.contMDiffAt (hU.mem_nhds hqu)).mdifferentiableAt (by simp))
theorem manifold_pullback_exterior_on {α : ManifoldOneForm E M}
(hα : ∀ c, ContDiffOn ℝ ∞ (chartOneForm α c) (extChartAt 𝓘(ℝ,E) c).target)
{g : N → M} {U : Set N} (hU : IsOpen U)
(hg : ContMDiffOn 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g U) {b : N} (hbU : b ∈ U) :
manifoldExteriorOneForm (E := F) (M := N)
(manifoldPullbackOneForm (E := E) (F := F) (fun _ => α) g 0) b =
manifoldPullbackTwoForm (E := E) (F := F) (fun _ => manifoldExteriorOneForm α) g 0 b := by
let y := extChartAt 𝓘(ℝ,F) b b
have hb := mem_extChartAt_source (I := 𝓘(ℝ,F)) b
have hy : y ∈ (extChartAt 𝓘(ℝ,F) b).target := (extChartAt 𝓘(ℝ,F) b).map_source hb
have he : (extChartAt 𝓘(ℝ,F) b).symm y=b := (extChartAt 𝓘(ℝ,F) b).left_inv hb
have hi := (contMDiffOn_extChartAt_symm (I := 𝓘(ℝ,F)) (n := ∞) b).contMDiffAt
((isOpen_extChartAt_target (I := 𝓘(ℝ,F)) b).mem_nhds hy)
have hgy : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g ((extChartAt 𝓘(ℝ,F) b).symm y) := by
rw [he]
exact hg.contMDiffAt (hU.mem_nhds hbU)
have hc : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y :=
hgy.comp y hi
have hu : ∀ᶠ z in 𝓝 y, (extChartAt 𝓘(ℝ,F) b).symm z ∈ U :=
hi.continuousAt.preimage_mem_nhds (hU.mem_nhds (he.symm ▸ hbU))
have hh : chartOneForm (E := F) (M := N)
(manifoldPullbackOneForm (E := E) (F := F) (fun _ => α) g 0) b =ᶠ[𝓝 y]
fun z => euclideanPullbackOneForm (fun _ => α) (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) (0,z) := by
filter_upwards [(isOpen_extChartAt_target (I := 𝓘(ℝ,F)) b).mem_nhds hy,hu] with z hz hzU
exact chartOneForm_manifoldPullback_at (α := fun _ => α) (b := b) (p := (0,z)) hz
((hg.contMDiffAt (hU.mem_nhds hzU)).mdifferentiableAt (by simp))
have hder : euclideanExteriorOneForm
(chartOneForm (E := F) (M := N)
(manifoldPullbackOneForm (E := E) (F := F) (fun _ => α) g 0) b) y =
(manifoldExteriorOneForm α (g b)).bilinearComp
(manifoldMapDifferential (E := E) (F := F) (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y)
(manifoldMapDifferential (E := E) (F := F) (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y) := by
calc
_ = euclideanExteriorOneForm
(fun z => euclideanPullbackOneForm (fun _ => α) (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) (0,z)) y := by
unfold euclideanExteriorOneForm
rw [hh.fderiv_eq]
_ = _ := by simpa only [Function.comp_apply,he] using euclidean_manifold_pullback_exterior hα hc
have hd := mfderiv_comp y (hgy.mdifferentiableAt (by simp))
(hi.mdifferentiableAt (by simp))
have hD : manifoldMapDifferential (E := E) (F := F)
(g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y =
(manifoldMapDifferential (E := E) (F := F) g b).comp
(chartDifferential (E := F) b b).inverse := by
change mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y = _
rw [hd,← chartDifferential_inverse hy]
change (manifoldMapDifferential (E := E) (F := F) g
((extChartAt 𝓘(ℝ,F) b).symm y)).comp
(chartDifferential (E := F) b ((extChartAt 𝓘(ℝ,F) b).symm y)).inverse = _
rw [he]
have H : (chartDifferential (E := F) b b).IsInvertible := by
convert! isInvertible_mfderiv_extChartAt (I := 𝓘(ℝ,F)) hb using 1
change (euclideanExteriorOneForm
(chartOneForm (E := F) (M := N)
(manifoldPullbackOneForm (E := E) (F := F) (fun _ => α) g 0) b) y).bilinearComp
(chartDifferential (E := F) b b) (chartDifferential (E := F) b b)=_
rw [hder,hD]
ext u v
change manifoldExteriorOneForm α (g b)
(mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) g b ((chartDifferential (E := F) b b).inverse (chartDifferential (E := F) b b u)))
(mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) g b ((chartDifferential (E := F) b b).inverse (chartDifferential (E := F) b b v)))=_
rw [H.inverse_apply_self,H.inverse_apply_self]
rfl
end
section
variable {P E F : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P]
[NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
{M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M]
def parameterEuclideanPullbackOneForm (α : P → ManifoldOneForm E M) (g : F → M)
(p : P × F) : F →L[ℝ] ℝ :=
(α p.1 (g p.2)).comp (mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) g p.2)
theorem parameterEuclideanPullbackOneForm_contDiffAt
{α : P → ManifoldOneForm E M} {g : F → M} {p : P × F}
(hα : ContDiffAt ℝ ∞
(fun q : P × E => chartOneForm (α q.1) (g p.2) q.2)
(p.1,extChartAt 𝓘(ℝ,E) (g p.2) (g p.2)))
(hg : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g p.2) :
ContDiffAt ℝ ∞ (parameterEuclideanPullbackOneForm α g) p := by
let c := g p.2
have hc : g p.2 ∈ (extChartAt 𝓘(ℝ,E) c).source := mem_extChartAt_source c
have hch := (contMDiffAt_extChartAt' (I := 𝓘(ℝ,E)) (n := ∞)
(x := c) (by simpa only [extChartAt_source] using hc)).comp p.2 hg
have hs : ContDiffAt ℝ ∞ (extChartAt 𝓘(ℝ,E) c ∘ g) p.2 := hch.contDiffAt
have hcoord : ContDiffAt ℝ ∞ (fun q : P × F => (q.1,extChartAt 𝓘(ℝ,E) c (g q.2))) p :=
contDiffAt_fst.prodMk (hs.comp p contDiffAt_snd)
have hform := hα.comp p hcoord
have hder := (hs.fderiv_right (m := ∞) (by simp)).comp p contDiffAt_snd
apply (hform.clm_comp hder).congr_of_eventuallyEq
have hg' := (contMDiffAt_iff_contMDiffAt_nhds (by simp : (1 : ℕ∞ω) ≠ ∞)).mp
(hg.of_le (by simp : (1 : ℕ∞ω) ≤ ∞))
have hc' := hg.continuousAt.preimage_mem_nhds
((isOpen_extChartAt_source (I := 𝓘(ℝ,E)) c).mem_nhds hc)
filter_upwards [continuousAt_snd.preimage_mem_nhds hg',
continuousAt_snd.preimage_mem_nhds hc'] with q hq hqc
exact euclideanPullbackOneForm_chart (α := fun _ => α q.1) (p := (0,q.2))
(hq.mdifferentiableAt (by simp)) hqc
end
section
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
{M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M]
theorem euclidean_manifold_pullback_exterior_at {α : ManifoldOneForm E M}
{g : F → M} {x : F} (hg : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g x)
(hα : ContDiffAt ℝ ∞ (chartOneForm α (g x)) (extChartAt 𝓘(ℝ,E) (g x) (g x))) :
euclideanExteriorOneForm (fun y => euclideanPullbackOneForm (fun _ => α) g (0,y)) x =
(manifoldExteriorOneForm α (g x)).bilinearComp
(manifoldMapDifferential (E := E) (F := F) g x)
(manifoldMapDifferential (E := E) (F := F) g x) := by
let c := g x
have hc : g x ∈ (extChartAt 𝓘(ℝ,E) c).source := mem_extChartAt_source c
have ho := contMDiffAt_extChartAt' (I := 𝓘(ℝ,E)) (n := ∞) (x := c)
(by simpa only [extChartAt_source] using hc)
have hgc : ContDiffAt ℝ ∞ (extChartAt 𝓘(ℝ,E) c ∘ g) x := (ho.comp x hg).contDiffAt
have he := euclidean_manifold_pullback_germ (α := α) hg hc
let A : F →L[ℝ] E := (chartDifferential c (g x)).comp
(manifoldMapDifferential (E := E) (F := F) g x)
have hf : HasFDerivAt (extChartAt 𝓘(ℝ,E) c ∘ g) A x :=
hasMFDerivAt_iff_hasFDerivAt.mp
((ho.mdifferentiableAt (by simp)).hasMFDerivAt.comp x
(hg.mdifferentiableAt (by simp)).hasMFDerivAt)
have hd := hf.fderiv
have hfd : fderiv ℝ (extChartAt 𝓘(ℝ,E) c ∘ g) x =
(chartDifferential (E := E) c (g x)).comp
(manifoldMapDifferential (E := E) (F := F) g x) := hd
calc
_ = euclideanExteriorOneForm
(staticPullbackOneForm (chartOneForm α c) (extChartAt 𝓘(ℝ,E) c ∘ g)) x := by
unfold euclideanExteriorOneForm
rw [he.fderiv_eq]
_ = _ := by
rw [staticPullbackOneForm_exterior
(g := extChartAt 𝓘(ℝ,E) c ∘ g) hα hgc,hfd]
rfl
end
section
variable {P E F : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P]
[NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
{M N : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M]
[TopologicalSpace N] [ChartedSpace F N] [IsManifold 𝓘(ℝ,F) ∞ N]
def parameterManifoldPullbackOneForm (α : P → ManifoldOneForm E M) (g : N → M)
(p : P) : ManifoldOneForm F N :=
manifoldPullbackOneForm (fun _ => α p) g 0
omit [NormedSpace ℝ P] in
theorem parameterManifoldPullback_chart_germ
(α : P → ManifoldOneForm E M) {g : N → M} (p : P) {b : N}
(hg : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g b) :
(fun q : P × F => chartOneForm (parameterManifoldPullbackOneForm α g q.1) b q.2)
=ᶠ[𝓝 (p,extChartAt 𝓘(ℝ,F) b b)]
parameterEuclideanPullbackOneForm α (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) := by
have hb := mem_extChartAt_source (I := 𝓘(ℝ,F)) b
have hy := (extChartAt 𝓘(ℝ,F) b).map_source hb
have he := (extChartAt 𝓘(ℝ,F) b).left_inv hb
have hi := (contMDiffOn_extChartAt_symm (I := 𝓘(ℝ,F)) (n := ∞) b).contMDiffAt
((isOpen_extChartAt_target (I := 𝓘(ℝ,F)) b).mem_nhds hy)
have hn := (contMDiffAt_iff_contMDiffAt_nhds (by simp : (1 : ℕ∞ω) ≠ ∞)).mp
(hg.of_le (by simp : (1 : ℕ∞ω) ≤ ∞))
have hn₁ : {x | ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) 1 g x} ∈
𝓝 ((extChartAt 𝓘(ℝ,F) b).symm (extChartAt 𝓘(ℝ,F) b b)) := by
rw [he]; exact hn
have hn' := (hi.continuousAt.comp (f := Prod.snd) (continuousAt_snd (p := (p,extChartAt 𝓘(ℝ,F) b b)))).preimage_mem_nhds hn₁
filter_upwards [hn',continuousAt_snd.preimage_mem_nhds
((isOpen_extChartAt_target (I := 𝓘(ℝ,F)) b).mem_nhds hy)] with q hq hqt
exact chartOneForm_manifoldPullback_at (α := fun _ => α q.1) (p := (0,q.2)) hqt
(hq.mdifferentiableAt (by simp))
theorem parameterManifoldPullbackOneForm_contDiffAt
{α : P → ManifoldOneForm E M} {g : N → M} {p : P} {b : N}
(hα : ContDiffAt ℝ ∞
(fun q : P × E => chartOneForm (α q.1) (g b) q.2)
(p,extChartAt 𝓘(ℝ,E) (g b) (g b)))
(hg : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g b) :
ContDiffAt ℝ ∞
(fun q : P × F => chartOneForm (parameterManifoldPullbackOneForm α g q.1) b q.2)
(p,extChartAt 𝓘(ℝ,F) b b) := by
let y := extChartAt 𝓘(ℝ,F) b b
have hb := mem_extChartAt_source (I := 𝓘(ℝ,F)) b
have hy : y∈(extChartAt 𝓘(ℝ,F) b).target := (extChartAt 𝓘(ℝ,F) b).map_source hb
have he : (extChartAt 𝓘(ℝ,F) b).symm y=b := (extChartAt 𝓘(ℝ,F) b).left_inv hb
have hi := (contMDiffOn_extChartAt_symm (I := 𝓘(ℝ,F)) (n := ∞) b).contMDiffAt
((isOpen_extChartAt_target (I := 𝓘(ℝ,F)) b).mem_nhds hy)
have hg' : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g ((extChartAt 𝓘(ℝ,F) b).symm y) := by
rw [he]; exact hg
have hα' : ContDiffAt ℝ ∞
(fun q : P × E => chartOneForm (α q.1) ((g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y) q.2)
(p,extChartAt 𝓘(ℝ,E) ((g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y)
((g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y)) := by
simpa only [Function.comp_apply,he] using hα
have hs := parameterEuclideanPullbackOneForm_contDiffAt
(α := α) (g := g ∘ (extChartAt 𝓘(ℝ,F) b).symm) (p := (p,y)) hα' (hg'.comp y hi)
exact hs.congr_of_eventuallyEq (parameterManifoldPullback_chart_germ α p hg)
theorem manifold_pullback_exterior_at {α : ManifoldOneForm E M}
{g : N → M} {b : N}
(hα : ContDiffAt ℝ ∞ (chartOneForm α (g b))
(extChartAt 𝓘(ℝ,E) (g b) (g b)))
(hg : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g b) :
manifoldExteriorOneForm (E := F) (M := N)
(manifoldPullbackOneForm (E := E) (F := F) (fun _ => α) g 0) b =
manifoldPullbackTwoForm (E := E) (F := F) (fun _ => manifoldExteriorOneForm α) g 0 b := by
let y := extChartAt 𝓘(ℝ,F) b b
have hb := mem_extChartAt_source (I := 𝓘(ℝ,F)) b
have hy : y ∈ (extChartAt 𝓘(ℝ,F) b).target := (extChartAt 𝓘(ℝ,F) b).map_source hb
have he : (extChartAt 𝓘(ℝ,F) b).symm y=b := (extChartAt 𝓘(ℝ,F) b).left_inv hb
have hi := (contMDiffOn_extChartAt_symm (I := 𝓘(ℝ,F)) (n := ∞) b).contMDiffAt
((isOpen_extChartAt_target (I := 𝓘(ℝ,F)) b).mem_nhds hy)
have hgy : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g ((extChartAt 𝓘(ℝ,F) b).symm y) := by
rw [he]
exact hg
have hc : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y :=
hgy.comp y hi
have hn := (contMDiffAt_iff_contMDiffAt_nhds (by simp : (1 : ℕ∞ω) ≠ ∞)).mp
(hgy.of_le (by simp : (1 : ℕ∞ω) ≤ ∞))
have hu := hi.continuousAt.preimage_mem_nhds hn
have hh : chartOneForm (E := F) (M := N)
(manifoldPullbackOneForm (E := E) (F := F) (fun _ => α) g 0) b =ᶠ[𝓝 y]
fun z => euclideanPullbackOneForm (fun _ => α) (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) (0,z) := by
filter_upwards [(isOpen_extChartAt_target (I := 𝓘(ℝ,F)) b).mem_nhds hy,hu] with z hz hzU
exact chartOneForm_manifoldPullback_at (α := fun _ => α) (b := b) (p := (0,z)) hz
(hzU.mdifferentiableAt (by simp))
have hder : euclideanExteriorOneForm
(chartOneForm (E := F) (M := N)
(manifoldPullbackOneForm (E := E) (F := F) (fun _ => α) g 0) b) y =
(manifoldExteriorOneForm α (g b)).bilinearComp
(manifoldMapDifferential (E := E) (F := F) (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y)
(manifoldMapDifferential (E := E) (F := F) (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y) := by
calc
_ = euclideanExteriorOneForm
(fun z => euclideanPullbackOneForm (fun _ => α) (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) (0,z)) y := by
unfold euclideanExteriorOneForm
rw [hh.fderiv_eq]
_ = _ := by
have hα' : ContDiffAt ℝ ∞
(chartOneForm α ((g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y))
(extChartAt 𝓘(ℝ,E) ((g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y)
((g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y)) := by
simpa only [Function.comp_apply,he] using hα
simpa only [Function.comp_apply,he] using euclidean_manifold_pullback_exterior_at hc hα'
have hd := mfderiv_comp y (hgy.mdifferentiableAt (by simp))
(hi.mdifferentiableAt (by simp))
have hD : manifoldMapDifferential (E := E) (F := F)
(g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y =
(manifoldMapDifferential (E := E) (F := F) g b).comp
(chartDifferential (E := F) b b).inverse := by
change mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) (g ∘ (extChartAt 𝓘(ℝ,F) b).symm) y = _
rw [hd,← chartDifferential_inverse hy]
change (manifoldMapDifferential (E := E) (F := F) g
((extChartAt 𝓘(ℝ,F) b).symm y)).comp
(chartDifferential (E := F) b ((extChartAt 𝓘(ℝ,F) b).symm y)).inverse = _
rw [he]
have H : (chartDifferential (E := F) b b).IsInvertible := by
convert! isInvertible_mfderiv_extChartAt (I := 𝓘(ℝ,F)) hb using 1
change (euclideanExteriorOneForm
(chartOneForm (E := F) (M := N)
(manifoldPullbackOneForm (E := E) (F := F) (fun _ => α) g 0) b) y).bilinearComp
(chartDifferential (E := F) b b) (chartDifferential (E := F) b b)=_
rw [hder,hD]
ext u v
change manifoldExteriorOneForm α (g b)
(mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) g b ((chartDifferential (E := F) b b).inverse (chartDifferential (E := F) b b u)))
(mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) g b ((chartDifferential (E := F) b b).inverse (chartDifferential (E := F) b b v)))=_
rw [H.inverse_apply_self,H.inverse_apply_self]
rfl
end
section
variable {P E V : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P]
[NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup V] [NormedSpace ℝ V]
{M : Type*} [TopologicalSpace M] [ChartedSpace E M]
instance (priority := 50) flatProduct_chartedSpace : ChartedSpace (E × V) (M × V) :=
prodChartedSpace E M V V
instance (priority := 50) flatProduct_manifold [IsManifold 𝓘(ℝ,E) ∞ M] :
IsManifold 𝓘(ℝ,E × V) ∞ (M × V) := by
rw [show 𝓘(ℝ,E × V)=(𝓘(ℝ,E)).prod 𝓘(ℝ,V) from modelWithCornersSelf_prod]
exact IsManifold.prod M V
omit [NormedAddCommGroup P] [NormedSpace ℝ P] in
theorem flatProduct_fst_smooth : ContMDiff 𝓘(ℝ,E × V) 𝓘(ℝ,E) ∞
(Prod.fst : M × V → M) := by
rw [show 𝓘(ℝ,E × V)=(𝓘(ℝ,E)).prod 𝓘(ℝ,V) from modelWithCornersSelf_prod]
exact contMDiff_fst
omit [NormedAddCommGroup P] [NormedSpace ℝ P] in
theorem flatProduct_snd_smooth : ContMDiff 𝓘(ℝ,E × V) 𝓘(ℝ,V) ∞
(Prod.snd : M × V → V) := by
rw [show 𝓘(ℝ,E × V)=(𝓘(ℝ,E)).prod 𝓘(ℝ,V) from modelWithCornersSelf_prod]
exact contMDiff_snd
omit [NormedAddCommGroup P] [NormedSpace ℝ P] in
theorem flatProduct_fst_derivative (z : M × V) :
mfderiv 𝓘(ℝ,E × V) 𝓘(ℝ,E) (Prod.fst : M × V → M) z=
ContinuousLinearMap.fst ℝ E V := by
rw [show 𝓘(ℝ,E × V)=(𝓘(ℝ,E)).prod 𝓘(ℝ,V) from modelWithCornersSelf_prod]
exact mfderiv_fst
def productHorizontalLift (Γ : V → ManifoldOneForm E M) : ManifoldOneForm (E × V) (M × V) :=
fun z => (Γ z.2 z.1).comp (ContinuousLinearMap.fst ℝ E V)
omit [NormedAddCommGroup P] [NormedSpace ℝ P] [TopologicalSpace M] [ChartedSpace E M] in
@[simp] theorem productHorizontalLift_apply (Γ : V → ManifoldOneForm E M)
(z : M × V) (v : E × V) : productHorizontalLift Γ z v=Γ z.2 z.1 v.1 := rfl
omit [NormedAddCommGroup P] [NormedSpace ℝ P] [TopologicalSpace M] [ChartedSpace E M] in
@[simp] theorem productHorizontalLift_vertical (Γ : V → ManifoldOneForm E M)
(z : M × V) (v : V) : productHorizontalLift Γ z (0,v)=0 := by
simp
variable [IsManifold 𝓘(ℝ,E) ∞ M]
omit [NormedAddCommGroup P] [NormedSpace ℝ P] in
theorem productHorizontalLift_chart (Γ : V → ManifoldOneForm E M) (c : M × V)
{y : E × V} (hy : y∈(extChartAt 𝓘(ℝ,E × V) c).target) :
chartOneForm (productHorizontalLift Γ) c y=
parameterEuclideanPullbackOneForm Γ
(Prod.fst ∘ (extChartAt 𝓘(ℝ,E × V) c).symm)
(((extChartAt 𝓘(ℝ,E × V) c).symm y).2,y) := by
let v := ((extChartAt 𝓘(ℝ,E × V) c).symm y).2
have he : chartOneForm (productHorizontalLift Γ) c y=
chartOneForm (manifoldPullbackOneForm (E := E) (F := E × V)
(fun _ => Γ v) (Prod.fst : M × V → M) 0) c y := by
simp only [chartOneForm,productHorizontalLift,manifoldPullbackOneForm,
manifoldMapDifferential,flatProduct_fst_derivative]
rfl
exact he.trans (chartOneForm_manifoldPullback (α := fun _ => Γ v)
(p := (0,y)) flatProduct_fst_smooth hy)
theorem productHorizontalLift_smooth {Γ : P × V → ManifoldOneForm E M}
(hΓ : SmoothOneFormFamily Γ) :
SmoothOneFormFamily (fun p => productHorizontalLift (fun v => Γ (p,v))) := by
intro c q hq
let g : E × V → M := Prod.fst ∘ (extChartAt 𝓘(ℝ,E × V) c).symm
let v : E × V → V := Prod.snd ∘ (extChartAt 𝓘(ℝ,E × V) c).symm
have hi := (contMDiffOn_extChartAt_symm (I := 𝓘(ℝ,E × V)) (n := ∞) c).contMDiffAt
((isOpen_extChartAt_target c).mem_nhds hq.2)
have hg : ContMDiffAt 𝓘(ℝ,E × V) 𝓘(ℝ,E) ∞ g q.2 :=
flatProduct_fst_smooth.contMDiffAt.comp q.2 hi
have hv : ContDiffAt ℝ ∞ v q.2 :=
(flatProduct_snd_smooth.contMDiffAt.comp q.2 hi).contDiffAt
have hx : ((q.1,v q.2),extChartAt 𝓘(ℝ,E) (g q.2) (g q.2))∈
univ ×ˢ (extChartAt 𝓘(ℝ,E) (g q.2)).target :=
⟨mem_univ _,(extChartAt 𝓘(ℝ,E) (g q.2)).map_source (mem_extChartAt_source _)⟩
have hs := parameterEuclideanPullbackOneForm_contDiffAt
(α := Γ) (p := ((q.1,v q.2),q.2))
((hΓ (g q.2)).contDiffAt ((isOpen_univ.prod (isOpen_extChartAt_target _)).mem_nhds hx)) hg
have hc := hs.comp q ((contDiffAt_fst.prodMk (hv.comp q contDiffAt_snd)).prodMk contDiffAt_snd)
apply (hc.congr_of_eventuallyEq ?_).contDiffWithinAt
filter_upwards [continuousAt_snd.preimage_mem_nhds
((isOpen_extChartAt_target (I := 𝓘(ℝ,E × V)) c).mem_nhds hq.2)] with z hz
exact productHorizontalLift_chart (fun w => Γ (z.1,w)) c hz
end
section
variable {F E V : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
[NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup V] [NormedSpace ℝ V]
{M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M]
def flatProductMap (g : F → M) : F × V → M × V := Prod.map g id
omit [IsManifold 𝓘(ℝ,E) ∞ M] in
theorem flatProductMap_smoothAt {g : F → M} {x : F × V}
(hg : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g x.1) :
ContMDiffAt 𝓘(ℝ,F × V) 𝓘(ℝ,E × V) ∞ (flatProductMap g) x := by
have h := hg.prodMap (contMDiffAt_id (I := 𝓘(ℝ,V)) (x := x.2))
simp +instances only [← modelWithCornersSelf_prod,chartedSpaceSelf_prod] at h
exact h
omit [IsManifold 𝓘(ℝ,E) ∞ M] in
theorem flatProductMap_derivative {g : F → M} {x : F × V}
(hg : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g x.1) :
mfderiv 𝓘(ℝ,F × V) 𝓘(ℝ,E × V) (flatProductMap g) x=
(mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) g x.1).prodMap (ContinuousLinearMap.id ℝ V) := by
have hf := (contDiff_fst : ContDiff ℝ ∞ (Prod.fst : F × V → F)).contMDiff.contMDiffAt (x := x)
have hs := (contDiff_snd : ContDiff ℝ ∞ (Prod.snd : F × V → V)).contMDiff.contMDiffAt (x := x)
rw [show 𝓘(ℝ,E × V)=(𝓘(ℝ,E)).prod 𝓘(ℝ,V) from modelWithCornersSelf_prod]
have hd := mfderiv_prodMk (I := 𝓘(ℝ,F × V)) (I' := 𝓘(ℝ,E)) (I'' := 𝓘(ℝ,V))
(f := g ∘ Prod.fst) (g := Prod.snd) (x := x)
((hg.comp x hf).mdifferentiableAt (by simp)) (hs.mdifferentiableAt (by simp))
change mfderiv 𝓘(ℝ,F × V) ((𝓘(ℝ,E)).prod 𝓘(ℝ,V))
(fun y : F × V => (g y.1,y.2)) x=_
apply hd.trans
erw [mfderiv_comp x (hg.mdifferentiableAt (by simp)) (hf.mdifferentiableAt (by simp)),
mfderiv_eq_fderiv,mfderiv_eq_fderiv,fderiv_fst,fderiv_snd]
rfl
omit [IsManifold 𝓘(ℝ,E) ∞ M] in
theorem productHorizontalLift_pullback_apply (Γ : V → ManifoldOneForm E M)
{g : F → M} {x : F × V} (hg : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g x.1)
(w : F × V) :
euclideanPullbackOneForm (fun _ => productHorizontalLift Γ) (flatProductMap g) (0,x) w=
euclideanPullbackOneForm (fun _ => Γ x.2) g (0,x.1) w.1 := by
change Γ x.2 (g x.1)
((mfderiv 𝓘(ℝ,F × V) 𝓘(ℝ,E × V) (flatProductMap g) x w).1)=
Γ x.2 (g x.1) (mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) g x.1 w.1)
rw [flatProductMap_derivative hg]
rfl
theorem productHorizontalLift_exterior_pullback {Γ : V → ManifoldOneForm E M}
(hΓ : SmoothOneFormFamily Γ) {g : F → M} {x : F × V}
(hg : ContMDiffAt 𝓘(ℝ,F) 𝓘(ℝ,E) ∞ g x.1) :
euclideanExteriorOneForm
(fun y => euclideanPullbackOneForm (fun _ => productHorizontalLift Γ) (flatProductMap g) (0,y)) x=
(manifoldExteriorOneForm (productHorizontalLift Γ) (flatProductMap g x)).bilinearComp
((mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) g x.1).prodMap (ContinuousLinearMap.id ℝ V))
((mfderiv 𝓘(ℝ,F) 𝓘(ℝ,E) g x.1).prodMap (ContinuousLinearMap.id ℝ V)) := by
have hfam : SmoothOneFormFamily (fun _ : ℝ => productHorizontalLift Γ) := by
apply productHorizontalLift_smooth (Γ := fun p : ℝ × V => Γ p.2)
intro c
have hm : ContDiffOn ℝ ∞ (fun p : (ℝ × V) × E => (p.1.2,p.2))
(univ ×ˢ (extChartAt 𝓘(ℝ,E) c).target) :=
((contDiff_snd.comp contDiff_fst).prodMk contDiff_snd).contDiffOn
exact (hΓ c).comp hm (fun _ hp => ⟨mem_univ _,hp.2⟩)
have he := euclidean_manifold_pullback_exterior_at (flatProductMap_smoothAt hg)
(hfam.spatial_smooth 0 (flatProductMap g x) (mem_extChartAt_target _))
simpa only [manifoldMapDifferential,flatProductMap_derivative hg] using he
end
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
theorem bilinearComp_equiv_isInvertible {A : E →L[ℝ] E →L[ℝ] ℝ}
(c : F ≃L[ℝ] E) (hA : A.IsInvertible) :
(A.bilinearComp (c : F →L[ℝ] E) (c : F →L[ℝ] E)).IsInvertible := by
obtain ⟨a,rfl⟩ := hA
refine ⟨(c.trans a).trans (c.symm.arrowCongr (ContinuousLinearEquiv.refl ℝ ℝ)),?_⟩
apply ContinuousLinearMap.ext
intro u
apply ContinuousLinearMap.ext
intro v
rfl
theorem bilinearComp_equiv_isInvertible_iff {A : E →L[ℝ] E →L[ℝ] ℝ}
(c : F ≃L[ℝ] E) :
(A.bilinearComp (c : F →L[ℝ] E) (c : F →L[ℝ] E)).IsInvertible ↔ A.IsInvertible := by
constructor
· intro h
have he : (A.bilinearComp (c : F →L[ℝ] E) (c : F →L[ℝ] E)).bilinearComp
(c.symm : E →L[ℝ] F) (c.symm : E →L[ℝ] F)=A := by
apply ContinuousLinearMap.ext
intro u
apply ContinuousLinearMap.ext
intro v
simp only [ContinuousLinearMap.bilinearComp_apply,ContinuousLinearEquiv.coe_coe,
ContinuousLinearEquiv.apply_symm_apply]
rw [← he]
exact bilinearComp_equiv_isInvertible c.symm h
· exact bilinearComp_equiv_isInvertible c
end PackingSufficiencySupport.Hamiltonian
end
end OAI
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