| import Mathlib | |
| import OAI.Geometry.Borsuk.Simplicial.UnitRadial | |
| namespace OAI | |
| open Set MeasureTheory | |
| namespace BorsukNine | |
| open scoped RealInnerProductSpace | |
| open SimplicialChains | |
| section HemisphereChart | |
| variable {E W : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] | |
| [NormedAddCommGroup W] [InnerProductSpace ℝ W] [CompleteSpace W] | |
| variable (w : E) (e : W →ₗᵢ[ℝ] E) | |
| noncomputable def hemisphereChart (v : W) : E := unitRadial (w + e v) | |
| noncomputable def hemisphereInverse (u : E) : W := ⟪w,u⟫⁻¹ • e.toContinuousLinearMap.adjoint u | |
| lemma adjoint_isometry_apply (v : W) : e.toContinuousLinearMap.adjoint (e v) = v := by | |
| have hh := congrArg (fun T : W →L[ℝ] W => T v) e.adjoint_comp_self | |
| exact hh | |
| lemma adjoint_normal (he : ∀ v, ⟪w,e v⟫ = 0) : e.toContinuousLinearMap.adjoint w = 0 := by | |
| apply ext_inner_right ℝ | |
| intro v | |
| rw [inner_zero_left,ContinuousLinearMap.adjoint_inner_left] | |
| exact he v | |
| lemma frame_decomposition (hw : ‖w‖ = 1) | |
| (he : ∀ v, ⟪w,e v⟫ = 0) (hes : ∀ u, ⟪w,u⟫ = 0 → ∃ v, e v = u) (u : E) : | |
| e (e.toContinuousLinearMap.adjoint u) = u - ⟪w,u⟫ • w := by | |
| obtain ⟨v,hv⟩ := hes (u - ⟪w,u⟫ • w) (by simp [inner_sub_right,inner_smul_right,hw]) | |
| have ha := congrArg e.toContinuousLinearMap.adjoint hv | |
| rw [adjoint_isometry_apply,map_sub,map_smul,adjoint_normal w e he,smul_zero,sub_zero] at ha | |
| rw [← ha,hv] | |
| omit [CompleteSpace E] [CompleteSpace W] in | |
| lemma hemisphere_add_ne_zero (hw : ‖w‖ = 1) (he : ∀ v, ⟪w,e v⟫ = 0) (v : W) : | |
| w + e v ≠ 0 := by | |
| intro hz | |
| have h := congrArg (fun u => ⟪w,u⟫) hz | |
| simp [inner_add_right,he,hw] at h | |
| omit [CompleteSpace E] [CompleteSpace W] in | |
| lemma hemisphereChart_contDiff (hw : ‖w‖ = 1) (he : ∀ v, ⟪w,e v⟫ = 0) : | |
| ContDiff ℝ (⊤ : ℕ∞) (hemisphereChart w e) := by | |
| have hc : ContDiff ℝ (⊤ : ℕ∞) (fun v => w + e v) := contDiff_const.add e.contDiff | |
| exact (hc.norm ℝ (hemisphere_add_ne_zero w e hw he)).inv | |
| (fun v => norm_ne_zero_iff.mpr (hemisphere_add_ne_zero w e hw he v)) |>.smul hc | |
| omit [CompleteSpace E] [CompleteSpace W] in | |
| lemma hemisphereChart_norm (hw : ‖w‖ = 1) (he : ∀ v, ⟪w,e v⟫ = 0) (v : W) : | |
| ‖hemisphereChart w e v‖ = 1 := unitRadial_norm (hemisphere_add_ne_zero w e hw he v) | |
| omit [CompleteSpace E] [CompleteSpace W] in | |
| lemma hemisphereChart_inner_pos (hw : ‖w‖ = 1) (he : ∀ v, ⟪w,e v⟫ = 0) (v : W) : | |
| 0 < ⟪w,hemisphereChart w e v⟫ := by | |
| simp only [hemisphereChart,unitRadial,inner_smul_right,inner_add_right, | |
| real_inner_self_eq_norm_sq,hw,one_pow,he,add_zero,mul_one] | |
| exact inv_pos.mpr (norm_pos_iff.mpr (hemisphere_add_ne_zero w e hw he v)) | |
| lemma hemisphereInverse_chart (hw : ‖w‖ = 1) (he : ∀ v, ⟪w,e v⟫ = 0) (v : W) : | |
| hemisphereInverse w e (hemisphereChart w e v) = v := by | |
| have hn : ‖w + e v‖ ≠ 0 := norm_ne_zero_iff.mpr (hemisphere_add_ne_zero w e hw he v) | |
| simp only [hemisphereInverse,hemisphereChart,unitRadial,inner_smul_right,inner_add_right, | |
| real_inner_self_eq_norm_sq,hw,one_pow,he,add_zero,mul_one,inv_inv,map_smul,map_add, | |
| adjoint_normal w e he,adjoint_isometry_apply,zero_add,smul_smul,mul_inv_cancel₀ hn,one_smul] | |
| lemma hemisphereChart_inverse (hw : ‖w‖ = 1) (he : ∀ v, ⟪w,e v⟫ = 0) | |
| (hes : ∀ u, ⟪w,u⟫ = 0 → ∃ v, e v = u) {u : E} (hu : ‖u‖ = 1) (hwu : 0 < ⟪w,u⟫) : | |
| hemisphereChart w e (hemisphereInverse w e u) = u := by | |
| have heq : w + e (hemisphereInverse w e u) = ⟪w,u⟫⁻¹ • u := by | |
| rw [hemisphereInverse,map_smul,frame_decomposition w e hw he hes,smul_sub,smul_smul, | |
| inv_mul_cancel₀ (ne_of_gt hwu),one_smul] | |
| abel | |
| rw [hemisphereChart,heq,unitRadial,norm_smul,Real.norm_eq_abs, | |
| abs_of_pos (inv_pos.mpr hwu),hu,mul_one,inv_inv,smul_smul, | |
| mul_inv_cancel₀ (ne_of_gt hwu),one_smul] | |
| lemma hemisphereInverse_continuousOn : | |
| ContinuousOn (hemisphereInverse w e) {u | 0 < ⟪w,u⟫} := | |
| ((continuous_const.inner continuous_id).continuousOn.inv₀ (fun _ h => ne_of_gt h)).smul | |
| e.toContinuousLinearMap.adjoint.continuous.continuousOn | |
| end HemisphereChart | |
| end BorsukNine | |
| end OAI | |