| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.PulledGradient |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
| namespace YauCounterexamples |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] {M : Type*} [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
|
|
| def bundleRankOne (g : SmoothMetric E M) (V : ∀ x : M, TangentSpace 𝓘(ℝ,E) x) (x : M) : |
| TangentSpace 𝓘(ℝ,E) x →L[ℝ] TangentSpace 𝓘(ℝ,E) x := |
| (g.inner x (V x)).smulRight (V x) |
|
|
| omit [FiniteDimensional ℝ E] in |
| lemma contMDiffAt_bundleRankOne (g : SmoothMetric E M) |
| {V : ∀ x : M, TangentSpace 𝓘(ℝ,E) x} {p : M} |
| (hv : ContMDiffAt 𝓘(ℝ,E) (𝓘(ℝ,E).prod 𝓘(ℝ,E)) ∞ |
| (fun x => TotalSpace.mk' E x (V x)) p) : |
| ContMDiffAt 𝓘(ℝ,E) (𝓘(ℝ,E).prod 𝓘(ℝ,E →L[ℝ] E)) ∞ |
| (fun x => TotalSpace.mk' (E →L[ℝ] E) x (bundleRankOne g V x)) p := by |
| rw [contMDiffAt_section p] at hv ⊢ |
| have hg := (contMDiffAt_localMetricFlat g p).clm_apply hv |
| have hs : ContDiff ℝ ∞ (fun z : (E →L[ℝ] ℝ) × E => z.1.smulRight z.2) := |
| contDiff_fst.smulRight contDiff_snd |
| have hprod := hg.prodMk hv |
| rw [← modelWithCornersSelf_prod, chartedSpaceSelf_prod] at hprod |
| have hh := hs.contMDiff.contMDiffAt.comp p hprod |
| apply hh.congr_of_eventuallyEq |
| filter_upwards [(trivializationAt E (TangentSpace 𝓘(ℝ,E)) p).open_baseSet.mem_nhds |
| (mem_baseSet_trivializationAt E _ p)] with x hx |
| simp only [Function.comp_def,hom_trivializationAt_apply] |
| rw [ContinuousLinearMap.inCoordinates_eq hx hx] |
| ext w |
| simp only [ContinuousLinearMap.comp_apply,ContinuousLinearEquiv.coe_coe, |
| bundleRankOne,ContinuousLinearMap.smulRight_apply,map_smul, |
| localMetricFlat_pairing g p x hx, |
| Trivialization.symm_apply_apply_mk _ hx, |
| Trivialization.continuousLinearEquivAt_apply, |
| Trivialization.continuousLinearEquivAt_symm_apply] |
|
|
| omit [FiniteDimensional ℝ E] in |
| lemma contMDiff_bundleId : |
| ContMDiff 𝓘(ℝ,E) (𝓘(ℝ,E).prod 𝓘(ℝ,E →L[ℝ] E)) ∞ |
| (fun x : M => TotalSpace.mk' (E →L[ℝ] E) x |
| (ContinuousLinearMap.id ℝ (TangentSpace 𝓘(ℝ,E) x))) := by |
| intro p |
| rw [contMDiffAt_section p] |
| apply (contMDiffAt_const (c := ContinuousLinearMap.id ℝ E)).congr_of_eventuallyEq |
| filter_upwards [(trivializationAt E (TangentSpace 𝓘(ℝ,E)) p).open_baseSet.mem_nhds |
| (mem_baseSet_trivializationAt E _ p)] with x hx |
| simp only [hom_trivializationAt_apply,ContinuousLinearMap.inCoordinates_eq hx hx] |
| ext w |
| simp |
|
|
| def bundleTransverse (g : SmoothMetric E M) (u : M → ℝ) (x : M) : |
| TangentSpace 𝓘(ℝ,E) x →L[ℝ] TangentSpace 𝓘(ℝ,E) x := |
| ContinuousLinearMap.id ℝ _ - |
| (g.inner x (metricGradient g u x) (metricGradient g u x))⁻¹ • |
| bundleRankOne g (metricGradient g u) x |
|
|
| lemma contMDiffAt_bundleTransverse (g : SmoothMetric E M) {u : M → ℝ} |
| (hu : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u) {p : M} |
| (hp : metricGradient g u p ≠ 0) : |
| ContMDiffAt 𝓘(ℝ,E) (𝓘(ℝ,E).prod 𝓘(ℝ,E →L[ℝ] E)) ∞ |
| (fun x => TotalSpace.mk' (E →L[ℝ] E) x (bundleTransverse g u x)) p := by |
| have hv := contMDiff_metricGradient g hu |
| have hq := (g.contMDiff p).clm_bundle_apply₂ (hv p) (hv p) |
| rw [contMDiffAt_totalSpace] at hq |
| exact (contMDiff_bundleId p).sub_section |
| ((hq.2.inv₀ (by simpa using (g.pos p _ hp).ne')).smul_section (contMDiffAt_bundleRankOne g (hv p))) |
|
|
| def supportedPinningTensor (g : SmoothMetric E M) (u v χ : M → ℝ) (x : M) : |
| TangentSpace 𝓘(ℝ,E) x →L[ℝ] TangentSpace 𝓘(ℝ,E) x := |
| let p := fun x => bundleTransverse g u x (metricGradient g v x) |
| χ x • (bundleRankOne g p x - (g.inner x (p x) (p x) /2) • bundleTransverse g u x) |
|
|
| theorem supportedPinningTensor_smooth (g : SmoothMetric E M) {u v χ : M → ℝ} |
| (hu : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u) |
| (hv : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ v) |
| (hχ : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ χ) {A : Set M} |
| (hA : IsOpen A) (hs : tsupport χ ⊆ A) (hn : ∀ x ∈ A, metricGradient g u x ≠ 0) : |
| ContMDiff 𝓘(ℝ,E) (𝓘(ℝ,E).prod 𝓘(ℝ,E →L[ℝ] E)) ∞ |
| (fun x => TotalSpace.mk' (E →L[ℝ] E) x (supportedPinningTensor g u v χ x)) := by |
| apply ContMDiffOn.smul_section_of_tsupport (hχ.contMDiffOn) hA hs |
| intro x hx |
| have hP := contMDiffAt_bundleTransverse g hu (hn x hx) |
| have hp := hP.clm_bundle_apply (contMDiff_metricGradient g hv x) |
| have hq := (g.contMDiff x).clm_bundle_apply₂ hp hp |
| rw [contMDiffAt_totalSpace] at hq |
| exact ((contMDiffAt_bundleRankOne g hp).sub_section |
| ((hq.2.div_const 2).smul_section hP)).contMDiffWithinAt |
| end YauCounterexamples |
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| end |
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| end OAI |
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