import OAI.Geometry.ConjugatePoints.Index import OAI.Geometry.ConjugatePoints.Curvature namespace OAI noncomputable section /-! # Geodesic flow, variation equations, and the Gauss lemma. -/ section open Set Filter Topology Asymptotics open scoped NNReal namespace ThreeManifold.ODEVariation variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] lemma remainder_bound {f : E → E} {D : E → E →L[ℝ] E} {u v : ℝ → E} {A : ℝ → E →L[ℝ] E} {x y : E} {a b K C L : ℝ} (hK : 0 < K) (hC : 0 ≤ C) (_hL : 0 ≤ L) (hu : ∀ t ∈ Icc a b, HasDerivAt u (f (u t)) t) (hv : ∀ t ∈ Icc a b, HasDerivAt v (f (v t)) t) (hA : ∀ t ∈ Icc a b, ∀ z, HasDerivAt (fun s => A s z) (D (u t) (A t z)) t) (hua : u a = x) (hva : v a = y) (hAa : A a = ContinuousLinearMap.id ℝ E) (hD : ∀ t ∈ Icc a b, ‖D (u t)‖ ≤ K) (hLip : ∀ t ∈ Icc a b, ‖v t - u t‖ ≤ L * ‖y-x‖) (hTaylor : ∀ t ∈ Icc a b, ‖f (v t) - f (u t) - D (u t) (v t-u t)‖ ≤ C * ‖v t-u t‖ ^ 2) : ∀ t ∈ Icc a b, ‖v t - u t - A t (y-x)‖ ≤ (C * L^2 / K * (Real.exp (K * (t-a))-1)) * ‖y-x‖^2 := by intro t ht let R : ℝ → E := fun s => v s-u s-A s (y-x) let R' : ℝ → E := fun s => f (v s)-f (u s)-D (u s) (A s (y-x)) have hd : ∀ s ∈ Icc a b, HasDerivAt R (R' s) s := fun s hs => ((hv s hs).sub (hu s hs)).sub (hA s hs (y-x)) have hb : ∀ s ∈ Ico a b, ‖R' s‖ ≤ K * ‖R s‖ + C*L^2*‖y-x‖^2 := by intro s hs have hsc := Ico_subset_Icc_self hs have he : R' s = D (u s) (R s) + (f (v s)-f (u s)-D (u s) (v s-u s)) := by simp only [R', R, map_sub] abel rw [he] calc _ ≤ ‖D (u s) (R s)‖ + ‖f (v s)-f (u s)-D (u s) (v s-u s)‖ := norm_add_le _ _ _ ≤ K * ‖R s‖ + C * ‖v s-u s‖^2 := add_le_add ((D (u s)).le_opNorm _ |>.trans (mul_le_mul_of_nonneg_right (hD s hsc) (norm_nonneg _))) (hTaylor s hsc) _ ≤ K * ‖R s‖ + C*L^2*‖y-x‖^2 := by have hh : ‖v s-u s‖^2 ≤ (L*‖y-x‖)^2 := pow_le_pow_left₀ (norm_nonneg _) (hLip s hsc) 2 nlinarith [mul_le_mul_of_nonneg_left hh hC] have hr := norm_le_gronwallBound_of_norm_deriv_right_le (f := R) (f' := R') (K := K) (ε := C*L^2*‖y-x‖^2) (δ := 0) (fun s hs => (hd s hs).continuousAt.continuousWithinAt) (fun s hs => (hd s (Ico_subset_Icc_self hs)).hasDerivWithinAt) (by simp [R,hua,hva,hAa]) hb t ht dsimp [gronwallBound] at hr rw [ite_eq_right hK.ne', zero_mul, zero_add] at hr change ‖R t‖ ≤ _ convert hr using 1 ; ring lemma hasFDerivAt_of_quadratic_error {F : E → E} {A : E →L[ℝ] E} {x : E} {Q : ℝ} (h : ∀ᶠ y in 𝓝 x, ‖F y-F x-A (y-x)‖ ≤ Q*‖y-x‖^2) : HasFDerivAt F A x := by rw [hasFDerivAt_iff_isLittleO] have hb : (fun y => F y-F x-A (y-x)) =O[𝓝 x] (fun y => ‖y-x‖^2) := by apply isBigO_iff.mpr refine ⟨Q,?_⟩ filter_upwards [h] with y hy simpa only [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg ‖y-x‖)] using hy exact hb.trans_isLittleO (isLittleO_pow_sub_sub x (by norm_num : 1 < (2 : ℕ))) lemma flow_hasFDerivAt {f : E → E} {D : E → E →L[ℝ] E} {X : E → ℝ → E} {A : ℝ → E →L[ℝ] E} {x : E} {a b K C L : ℝ} (hK : 0 < K) (hC : 0 ≤ C) (hL : 0 ≤ L) (hx : ∀ t ∈ Icc a b, HasDerivAt (X x) (f (X x t)) t) (hA : ∀ t ∈ Icc a b, ∀ z, HasDerivAt (fun s => A s z) (D (X x t) (A t z)) t) (hxa : X x a = x) (hAa : A a = ContinuousLinearMap.id ℝ E) (hD : ∀ t ∈ Icc a b, ‖D (X x t)‖ ≤ K) (hnear : ∀ᶠ y in 𝓝 x, X y a = y ∧ (∀ t ∈ Icc a b, HasDerivAt (X y) (f (X y t)) t) ∧ (∀ t ∈ Icc a b, ‖X y t-X x t‖ ≤ L*‖y-x‖) ∧ (∀ t ∈ Icc a b, ‖f (X y t)-f (X x t)-D (X x t) (X y t-X x t)‖ ≤ C*‖X y t-X x t‖^2)) {t : ℝ} (ht : t ∈ Icc a b) : HasFDerivAt (fun y => X y t) (A t) x := by apply hasFDerivAt_of_quadratic_error (Q := C*L^2/K*(Real.exp (K*(t-a))-1)) filter_upwards [hnear] with y hy exact remainder_bound hK hC hL hx hy.2.1 hA hxa hy.1 hAa hD hy.2.2.1 hy.2.2.2 t ht lemma quadratic_taylor {f : E → E} {S : Set E} {C : ℝ≥0} (hS : Convex ℝ S) (hf : ∀ z ∈ S, DifferentiableAt ℝ f z) (hD : LipschitzOnWith C (fderiv ℝ f) S) {x y : E} (hx : x ∈ S) (hy : y ∈ S) : ‖f y-f x-fderiv ℝ f x (y-x)‖ ≤ C*‖y-x‖^2 := by have hseg := hS.segment_subset hx hy have hd : ∀ z ∈ segment ℝ x y, ‖fderiv ℝ f z-fderiv ℝ f x‖ ≤ C*‖y-x‖ := by intro z hz exact (hD.norm_sub_le (hseg hz) hx).trans (mul_le_mul_of_nonneg_left (norm_sub_le_of_mem_segment hz) C.coe_nonneg) have hh := (convex_segment x y).norm_image_sub_le_of_norm_fderiv_le' (fun z hz => hf z (hseg hz)) hd (left_mem_segment ℝ x y) (right_mem_segment ℝ x y) simpa only [pow_two,mul_assoc] using hh lemma local_bounds {f : E → E} {x : E} (hf : ContDiffAt ℝ 2 f x) : ∃ r > (0 : ℝ), ∃ K > (0 : ℝ), ∃ C : ℝ≥0, (∀ z ∈ Metric.ball x r, DifferentiableAt ℝ f z) ∧ (∀ z ∈ Metric.ball x r, ‖fderiv ℝ f z‖ ≤ K) ∧ (∀ z ∈ Metric.ball x r, ∀ w ∈ Metric.ball x r, ‖f w-f z-fderiv ℝ f z (w-z)‖ ≤ C*‖w-z‖^2) := by have hD : ContDiffAt ℝ 1 (fderiv ℝ f) x := hf.fderiv_right (by norm_num) obtain ⟨C,S,hS,hLip⟩ := hD.exists_lipschitzOnWith have hdiff : ∀ᶠ z in 𝓝 x, DifferentiableAt ℝ f z := by filter_upwards [hf.eventually (by norm_num)] with z hz exact hz.differentiableAt (by norm_num) have hbound : ∀ᶠ z in 𝓝 x, ‖fderiv ℝ f z‖ < ‖fderiv ℝ f x‖+1 := hD.continuousAt.norm.eventually_lt_const (by linarith) obtain ⟨r,hr,hrs⟩ := Metric.mem_nhds_iff.mp (inter_mem hS (inter_mem hdiff hbound)) refine ⟨r,hr,‖fderiv ℝ f x‖+1,by positivity,C,?_,?_,?_⟩ · intro z hz; exact (hrs hz).2.1 · intro z hz; exact (hrs hz).2.2.le · intro z hz w hw exact quadratic_taylor (convex_ball x r) (fun z hz => (hrs hz).2.1) (hLip.mono (fun z hz => (hrs hz).1)) hz hw end ThreeManifold.ODEVariation end section open Set Filter Topology Metric open scoped NNReal namespace ThreeManifold.ODEVariation variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] def variationVector (f : E → E) (z : E × (E →L[ℝ] E)) : E × (E →L[ℝ] E) := (f z.1, (fderiv ℝ f z.1).comp z.2) lemma variationVector_smoothAt {f : E → E} {x : E} (hf : ContDiffAt ℝ 2 f x) (A : E →L[ℝ] E) : ContDiffAt ℝ 1 (variationVector f) (x,A) := by let _ : CompleteSpace E := inferInstance exact ((hf.of_le (by norm_num)).comp (x,A) contDiffAt_fst).prodMk (((hf.fderiv_right (m := 1) (by norm_num)).comp (x,A) contDiffAt_fst).clm_comp contDiffAt_snd) lemma exists_variation_flow {f : E → E} {x : E} (hf : ContDiffAt ℝ 2 f x) : ∃ r > (0 : ℝ), ∃ ε > (0 : ℝ), ∃ X : E → ℝ → E, ∃ A : E → ℝ → E →L[ℝ] E, ∃ L : ℝ≥0, (∀ y ∈ closedBall x r, X y 0 = y ∧ A y 0 = ContinuousLinearMap.id ℝ E) ∧ (∀ y ∈ closedBall x r, ∀ t ∈ Icc (-ε) ε, HasDerivAt (X y) (f (X y t)) t ∧ HasDerivAt (A y) ((fderiv ℝ f (X y t)).comp (A y t)) t) ∧ (∀ t ∈ Icc (-ε) ε, ∀ y ∈ closedBall x r, ∀ z ∈ closedBall x r, ‖X y t-X z t‖ ≤ L*‖y-z‖ ∧ ‖A y t-A z t‖ ≤ L*‖y-z‖) ∧ ContinuousOn (fun yt : E × ℝ => (X yt.1 yt.2,A yt.1 yt.2)) (closedBall x r ×ˢ Icc (-ε) ε) := by let z₀ : E × (E →L[ℝ] E) := (x,ContinuousLinearMap.id ℝ E) obtain ⟨ε,hε,a,r,B,K,hr,hpl⟩ := IsPicardLindelof.of_contDiffAt_one (variationVector_smoothAt hf (ContinuousLinearMap.id ℝ E)) obtain ⟨Y,hY,L,hLip⟩ := (hpl 0).exists_forall_mem_closedBall_eq_hasDerivWithinAt_lipschitzOnWith have hcont : ContinuousOn (Function.uncurry Y) (closedBall z₀ r ×ˢ Icc (0-ε) (0+ε)) := by apply continuousOn_prod_of_continuousOn_lipschitzOnWith _ L _ hLip intro z hz exact HasDerivWithinAt.continuousOn (hY z hz).2 let ι : E → E × (E →L[ℝ] E) := fun y => (y,ContinuousLinearMap.id ℝ E) have hi : LipschitzWith 1 ι := by apply LipschitzWith.of_dist_le_mul intro z w simp [ι,Prod.dist_eq] have hi_mem : ∀ y ∈ closedBall x r, ι y ∈ closedBall z₀ r := by intro y hy change dist y x ≤ (r : ℝ) at hy change dist (ι y) z₀ ≤ (r : ℝ) simpa only [ι,z₀,Prod.dist_eq,dist_self,max_eq_left dist_nonneg] using hy have htime : Icc (-(ε/2)) (ε/2) ⊆ Icc (0-ε) (0+ε) := by rintro t ⟨ht1,ht2⟩; constructor <;> linarith have htime_nhds : ∀ t ∈ Icc (-(ε/2)) (ε/2), Icc (0-ε) (0+ε) ∈ 𝓝 t := by rintro t ⟨ht1,ht2⟩ apply Icc_mem_nhds <;> linarith refine ⟨r,hr,ε/2,half_pos hε,(fun y t => (Y (ι y) t).1), (fun y t => (Y (ι y) t).2),L,?_,?_,?_,?_⟩ · intro y hy have hy0 := (hY (ι y) (hi_mem y hy)).1 exact ⟨congrArg Prod.fst hy0,congrArg Prod.snd hy0⟩ · intro y hy t ht have hd := ((hY (ι y) (hi_mem y hy)).2 t (htime ht)).hasDerivAt (htime_nhds t ht) exact ⟨hd.fst,hd.snd⟩ · intro t ht y hy z hz have hh := (hLip t (htime ht)).norm_sub_le (hi_mem y hy) (hi_mem z hz) have he : ‖ι y-ι z‖ = ‖y-z‖ := by change max ‖y-z‖ ‖(ContinuousLinearMap.id ℝ E)-(ContinuousLinearMap.id ℝ E)‖ = ‖y-z‖ simp rw [he] at hh exact ⟨(norm_fst_le _).trans hh,(norm_snd_le _).trans hh⟩ · change ContinuousOn (Function.uncurry Y ∘ (fun yt : E × ℝ => (ι yt.1,yt.2))) _ apply hcont.comp ((hi.continuous.comp continuous_fst).prodMk continuous_snd).continuousOn intro yt hyt exact ⟨hi_mem yt.1 hyt.1,htime hyt.2⟩ lemma flow_derivative_on {f : E → E} {X : E → ℝ → E} {A : E → ℝ → E →L[ℝ] E} {S : Set E} {ε K C L : ℝ} (hS : IsOpen S) (hK : 0 < K) (hC : 0 ≤ C) (hL : 0 ≤ L) (h0 : ∀ y ∈ S, X y 0 = y ∧ A y 0 = ContinuousLinearMap.id ℝ E) (hODE : ∀ y ∈ S, ∀ t ∈ Icc (-ε) ε, HasDerivAt (X y) (f (X y t)) t ∧ HasDerivAt (A y) ((fderiv ℝ f (X y t)).comp (A y t)) t) (hD : ∀ y ∈ S, ∀ t ∈ Icc (-ε) ε, ‖fderiv ℝ f (X y t)‖ ≤ K) (hLip : ∀ y ∈ S, ∀ z ∈ S, ∀ t ∈ Icc (-ε) ε, ‖X z t-X y t‖ ≤ L*‖z-y‖) (hTaylor : ∀ y ∈ S, ∀ z ∈ S, ∀ t ∈ Icc (-ε) ε, ‖f (X z t)-f (X y t)-fderiv ℝ f (X y t) (X z t-X y t)‖ ≤ C*‖X z t-X y t‖^2) {y : E} (hy : y ∈ S) {t : ℝ} (ht : t ∈ Icc (-ε) ε) : HasFDerivAt (fun z => X z t) (A y t) y := by let _ : CompleteSpace E := inferInstance have hε : 0 ≤ ε := by have := ht.1; have := ht.2; linarith have hp : Icc (0 : ℝ) ε ⊆ Icc (-ε) ε := by rintro s ⟨hs0,hsε⟩; exact ⟨by linarith,hsε⟩ have hm : ∀ s ∈ Icc (0 : ℝ) ε, -s ∈ Icc (-ε) ε := by rintro s ⟨hs0,hsε⟩; constructor <;> linarith by_cases ht0 : 0 ≤ t · apply flow_hasFDerivAt hK hC hL (b := ε) (fun s hs => (hODE y hy s (hp hs)).1) (fun s hs z => by simpa only [ContinuousLinearMap.comp_apply,map_zero,add_zero] using (hODE y hy s (hp hs)).2.clm_apply (hasDerivAt_const s z)) (h0 y hy).1 (h0 y hy).2 (fun s hs => hD y hy s (hp hs)) (ht := ⟨ht0,ht.2⟩) filter_upwards [hS.mem_nhds hy] with z hz exact ⟨(h0 z hz).1,(fun s hs => (hODE z hz s (hp hs)).1), (fun s hs => hLip y hy z hz s (hp hs)),(fun s hs => hTaylor y hy z hz s (hp hs))⟩ · have hxrev : ∀ z ∈ S, ∀ s ∈ Icc (0 : ℝ) ε, HasDerivAt (fun r => X z (-r)) (-(f (X z (-s)))) s := by intro z hz s hs simpa only [Function.comp_def,neg_one_smul] using ((hODE z hz (-s) (hm s hs)).1.scomp s (hasDerivAt_id s).neg) have hArev : ∀ s ∈ Icc (0 : ℝ) ε, ∀ v, HasDerivAt (fun r => A y (-r) v) ((-fderiv ℝ f (X y (-s))) (A y (-s) v)) s := by intro s hs v have hh := ((hODE y hy (-s) (hm s hs)).2.clm_apply (hasDerivAt_const (-s) v)).scomp s (hasDerivAt_id s).neg simpa only [Function.comp_def,ContinuousLinearMap.comp_apply,_root_.neg_apply, neg_one_smul,map_zero,add_zero] using hh have hh : HasFDerivAt (fun z => X z (-(-t))) (A y (-(-t))) y := by apply flow_hasFDerivAt (f := fun w => -f w) (D := fun w => -fderiv ℝ f w) (X := fun z s => X z (-s)) (A := fun s => A y (-s)) hK hC hL (b := ε) (hxrev y hy) hArev (by simpa using (h0 y hy).1) (by simpa using (h0 y hy).2) (fun s hs => by simpa only [norm_neg] using hD y hy (-s) (hm s hs)) (ht := ⟨by linarith,by linarith [ht.1]⟩) filter_upwards [hS.mem_nhds hy] with z hz refine ⟨by simpa using (h0 z hz).1,hxrev z hz,(fun s hs => hLip y hy z hz (-s) (hm s hs)),?_⟩ intro s hs have he : (-f (X z (-s)))-(-f (X y (-s)))-(-fderiv ℝ f (X y (-s))) (X z (-s)-X y (-s)) = -(f (X z (-s))-f (X y (-s))-fderiv ℝ f (X y (-s)) (X z (-s)-X y (-s))) := by simp only [_root_.neg_apply]; abel rw [he,norm_neg] exact hTaylor y hy z hz (-s) (hm s hs) simpa only [neg_neg] using hh lemma shrink_flow_image {x : E} {r ε R : ℝ} {X : E → ℝ → E} (hr : 0 < r) (hε : 0 < ε) (hR : 0 < R) (h0 : X x 0 = x) (hc : ContinuousOn (Function.uncurry X) (closedBall x r ×ˢ Icc (-ε) ε)) : ∃ ρ > (0 : ℝ), ∃ δ > (0 : ℝ), ∀ y ∈ ball x ρ, ∀ t ∈ Icc (-δ) δ, X y t ∈ ball x R ∧ y ∈ closedBall x r ∧ t ∈ Icc (-ε) ε := by let _ : NormedSpace ℝ E := inferInstance let _ : CompleteSpace E := inferInstance have hn : closedBall x r ×ˢ Icc (-ε) ε ∈ 𝓝 (x,(0 : ℝ)) := by rw [nhds_prod_eq] exact prod_mem_prod (closedBall_mem_nhds x hr) (Icc_mem_nhds (by linarith) hε) have hxc : ContinuousAt (Function.uncurry X) (x,(0 : ℝ)) := hc.continuousAt hn have hm : (Function.uncurry X) ⁻¹' ball x R ∈ 𝓝 (x,(0 : ℝ)) := by apply hxc.preimage_mem_nhds apply isOpen_ball.mem_nhds simpa only [Function.uncurry_apply_pair,h0] using mem_ball_self hR have hd : ball x r ×ˢ Ioo (-ε) ε ∈ 𝓝 (x,(0 : ℝ)) := by rw [nhds_prod_eq] exact prod_mem_prod (ball_mem_nhds x hr) (Ioo_mem_nhds (by linarith) hε) obtain ⟨η,hη,hsub⟩ := Metric.mem_nhds_iff.mp (inter_mem hm hd) refine ⟨η/2,half_pos hη,η/2,half_pos hη,?_⟩ intro y hy t ht have hp : (y,t) ∈ ball (x,(0 : ℝ)) η := by rw [mem_ball,Prod.dist_eq,max_lt_iff] constructor · exact lt_trans hy (half_lt_self hη) · rw [Real.dist_eq,sub_zero,abs_lt] constructor <;> linarith [ht.1,ht.2] exact ⟨(hsub hp).1,ball_subset_closedBall (hsub hp).2.1, Ioo_subset_Icc_self (hsub hp).2.2⟩ lemma exists_contDiff_variation_flow {f : E → E} {x : E} (hf : ContDiffAt ℝ 2 f x) : ∃ r > (0 : ℝ), ∃ ε > (0 : ℝ), ∃ X : E → ℝ → E, (∀ y ∈ ball x r, X y 0 = y) ∧ (∀ y ∈ ball x r, ∀ t ∈ Ioo (-ε) ε, HasDerivAt (X y) (f (X y t)) t) ∧ ContDiffOn ℝ 1 (Function.uncurry X) (ball x r ×ˢ Ioo (-ε) ε) ∧ ∃ A : E → ℝ → E →L[ℝ] E, (∀ y ∈ ball x r, A y 0 = ContinuousLinearMap.id ℝ E) ∧ (∀ y ∈ ball x r, ∀ t ∈ Ioo (-ε) ε, HasFDerivAt (fun z => X z t) (A y t) y) ∧ (∀ y ∈ ball x r, ∀ t ∈ Ioo (-ε) ε, HasDerivAt (A y) ((fderiv ℝ f (X y t)).comp (A y t)) t) := by obtain ⟨R,hR,K,hK,C,hfd,hDf,hTaylor⟩ := local_bounds hf obtain ⟨r,hr,ε,hε,X,A,L,h0,hODE,hLip,hcont⟩ := exists_variation_flow hf have h0x : X x 0 = x := (h0 x (mem_closedBall_self hr.le)).1 obtain ⟨ρ,hρ,δ,hδ,hsmall⟩ := shrink_flow_image hr hε hR h0x hcont.fst have hz : (0 : ℝ) ∈ Icc (-δ) δ := ⟨by linarith,hδ.le⟩ have hinit : ∀ y ∈ ball x ρ, y ∈ closedBall x r := fun y hy => (hsmall y hy 0 hz).2.1 have htime : Icc (-δ) δ ⊆ Icc (-ε) ε := fun t ht => (hsmall x (mem_ball_self hρ) t ht).2.2 have hdf : ∀ y ∈ ball x ρ, ∀ t ∈ Icc (-δ) δ, HasFDerivAt (fun z => X z t) (A y t) y := by intro y hy t ht apply flow_derivative_on isOpen_ball hK C.coe_nonneg L.coe_nonneg (fun y hy => h0 y (hinit y hy)) (fun y hy t ht => hODE y (hinit y hy) t (htime ht)) (fun y hy t ht => hDf _ (hsmall y hy t ht).1) (fun y hy z hz t ht => (hLip t (htime ht) z (hinit z hz) y (hinit y hy)).1) (fun y hy z hz t ht => hTaylor _ (hsmall y hy t ht).1 _ (hsmall z hz t ht).1) hy ht let O := ball x ρ ×ˢ Ioo (-δ) δ have hO : IsOpen O := isOpen_ball.prod isOpen_Ioo have hcont' : ContinuousOn (fun yt : E × ℝ => (X yt.1 yt.2,A yt.1 yt.2)) O := hcont.mono (fun p hp => ⟨hinit p.1 hp.1,htime (Ioo_subset_Icc_self hp.2)⟩) have hX : ContinuousOn (Function.uncurry X) O := hcont'.fst have hA : ContinuousOn (Function.uncurry A) O := hcont'.snd have hfX : ContinuousOn (fun p : E × ℝ => f (X p.1 p.2)) O := by intro p hp exact (hfd _ (hsmall p.1 hp.1 p.2 (Ioo_subset_Icc_self hp.2)).1).continuousAt.comp_continuousWithinAt (f := Function.uncurry X) (hX p hp) let B : E × ℝ → ℝ →L[ℝ] E := fun p => ContinuousLinearMap.toSpanSingleton ℝ (f (X p.1 p.2)) have hB : ContinuousOn B O := (ContinuousLinearMap.toSpanSingletonCLE (𝕜 := ℝ) (E := E)).continuous.comp_continuousOn hfX let D : E × ℝ → (E × ℝ) →L[ℝ] E := fun p => (A p.1 p.2).coprod (B p) have hd : ∀ p ∈ O, HasFDerivAt (Function.uncurry X) (D p) p := by intro p hp apply HasStrictFDerivAt.hasFDerivAt apply hasStrictFDerivAt_uncurry_coprod (f₁ := A) (f₂ := fun y t => B (y,t)) · filter_upwards [hO.mem_nhds hp] with q hq exact hdf q.1 hq.1 q.2 (Ioo_subset_Icc_self hq.2) · filter_upwards [hO.mem_nhds hp] with q hq exact (hODE q.1 (hinit q.1 hq.1) q.2 (htime (Ioo_subset_Icc_self hq.2))).1.hasFDerivAt · exact hA.continuousAt (hO.mem_nhds hp) · exact hB.continuousAt (hO.mem_nhds hp) have hD : ContinuousOn D O := by have hh := (hA.clm_comp (continuousOn_const (c := ContinuousLinearMap.fst ℝ E ℝ))).add (hB.clm_comp (continuousOn_const (c := ContinuousLinearMap.snd ℝ E ℝ))) exact hh refine ⟨ρ,hρ,δ,hδ,X,(fun y hy => (h0 y (hinit y hy)).1), (fun y hy t ht => (hODE y (hinit y hy) t (htime (Ioo_subset_Icc_self ht))).1),?_⟩ constructor · apply (contDiffOn_succ_iff_fderiv_of_isOpen (n := 0) hO).mpr refine ⟨fun p hp => (hd p hp).differentiableAt.differentiableWithinAt,by simp,?_⟩ rw [contDiffOn_zero] exact hD.congr (fun p hp => (hd p hp).fderiv) · exact ⟨A,(fun y hy => (h0 y (hinit y hy)).2), (fun y hy t ht => hdf y hy t (Ioo_subset_Icc_self ht)), (fun y hy t ht => (hODE y (hinit y hy) t (htime (Ioo_subset_Icc_self ht))).2)⟩ lemma exists_contDiff_flow {f : E → E} {x : E} (hf : ContDiffAt ℝ 2 f x) : ∃ r > (0 : ℝ), ∃ ε > (0 : ℝ), ∃ X : E → ℝ → E, (∀ y ∈ ball x r, X y 0 = y) ∧ (∀ y ∈ ball x r, ∀ t ∈ Ioo (-ε) ε, HasDerivAt (X y) (f (X y t)) t) ∧ ContDiffOn ℝ 1 (Function.uncurry X) (ball x r ×ˢ Ioo (-ε) ε) := by obtain ⟨r,hr,ε,hε,X,h0,hd,hs,_⟩ := exists_contDiff_variation_flow hf exact ⟨r,hr,ε,hε,X,h0,hd,hs⟩ end ThreeManifold.ODEVariation end section open Set Filter Topology Bundle Manifold IsManifold Matrix open scoped ContDiff Matrix.Norms.Elementwise namespace ThreeManifold.ODEVariation variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] lemma solution_smooth {f : E → E} {S : Set E} {U : Set ℝ} {x : ℝ → E} (hf : ContDiffOn ℝ ∞ f S) (hU : IsOpen U) (hx : MapsTo x U S) (hd : ∀ t ∈ U, HasDerivAt x (f (x t)) t) : ContDiffOn ℝ ∞ x U := by rw [contDiffOn_infty] intro n induction n with | zero => exact (contDiffOn_zero).2 (fun t ht => (hd t ht).continuousAt.continuousWithinAt) | succ n hn => rw [Nat.cast_add,Nat.cast_one,contDiffOn_succ_iff_deriv_of_isOpen hU] refine ⟨fun t ht => (hd t ht).differentiableAt.differentiableWithinAt,by simp,?_⟩ exact ((contDiffOn_infty.mp hf n).comp hn hx).congr (fun t ht => (hd t ht).deriv) end ThreeManifold.ODEVariation namespace ThreeManifold.RiemannianCoordinates variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] def geodesicPhase (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3) (z : E3 × E3) : E3 × E3 := (z.2,-connectionTerm g c z.1 z.2 z.2) lemma geodesicPhase_smoothAt (g : SmoothMetric M) {c : OpenPartialHomeomorph M E3} (hc : c ∈ atlas E3 M) {z : E3 × E3} (hz : z.1 ∈ c.target) : ContDiffAt ℝ ∞ (geodesicPhase g c) z := by apply contDiffAt_snd.prodMk apply ContDiffAt.neg apply contDiffAt_pi.mpr intro i apply ContDiffAt.sum intro j _ apply ContDiffAt.sum intro k _ exact (((christoffel_smoothAt g hc hz i j k).comp z contDiffAt_fst).mul (contDiffAt_pi.mp contDiffAt_snd j)).mul (contDiffAt_pi.mp contDiffAt_snd k) lemma chart_geodesic (g : SmoothMetric M) {c : OpenPartialHomeomorph M E3} (hc : c ∈ atlas E3 M) {U : Set ℝ} (hU : IsOpen U) {z : ℝ → E3 × E3} (hz : ∀ t ∈ U, (z t).1 ∈ c.target) (hd : ∀ t ∈ U, HasDerivAt z (geodesicPhase g c (z t)) t) : IsGeodesicOn g (fun t => c.symm (z t).1) U := by have hzs : ContDiffOn ℝ ∞ z U := ODEVariation.solution_smooth (fun x hx => (geodesicPhase_smoothAt g hc hx).contDiffWithinAt) hU hz hd have hxs : ContDiffOn ℝ ∞ (fun t => (z t).1) U := hzs.fst have hvs : ContDiffOn ℝ ∞ (fun t => (z t).2) U := hzs.snd have hγ : ContMDiffOn 𝓘(ℝ,ℝ) I3 ∞ (fun t => c.symm (z t).1) U := by intro t ht exact ((contMDiffAt_symm_of_mem_maximalAtlas (subset_maximalAtlas hc) (hz t ht)).comp t (contMDiffAt_iff_contDiffAt.mpr (hxs.contDiffAt (hU.mem_nhds ht)))).contMDiffWithinAt refine ⟨hγ,?_⟩ intro d hdc t ht hdt have hdx : ∀ r ∈ U, HasDerivAt (fun s => (z s).1) (z r).2 r := fun r hr => (hd r hr).fst have hdv : ∀ r ∈ U, HasDerivAt (fun s => (z s).2) (-connectionTerm g c (z r).1 (z r).2 (z r).2) r := fun r hr => (hd r hr).snd have heq : (fun r => deriv (fun s => (z s).1) r) =ᶠ[𝓝 t] fun r => (z r).2 := by filter_upwards [hU.mem_nhds ht] with r hr exact (hdx r hr).deriv have hacc : alongDerivative g c (fun s => (z s).1) (fun s => (z s).2) t = 0 := by rw [alongDerivative,(hdx t ht).deriv,(hdv t ht).deriv,neg_add_cancel] have hchange := along_change g hc hdc (hdx t ht).differentiableAt (hdv t ht).differentiableAt (hz t ht) hdt dsimp only at hchange rw [hacc,map_zero] at hchange have hdtrans : (fun r => fderiv ℝ (d ∘ c.symm) ((z r).1) ((z r).2)) =ᶠ[𝓝 t] deriv ((d ∘ c.symm) ∘ (fun r => (z r).1)) := by have hnh := (hγ.continuousOn.continuousAt (hU.mem_nhds ht)).preimage_mem_nhds (d.open_source.mem_nhds hdt) filter_upwards [hU.mem_nhds ht,hnh] with r hr hdr exact ((transition_smoothAt hc hdc (hz r hr) hdr).differentiableAt (by simp)).hasFDerivAt |>.comp_hasDerivAt r (hdx r hr) |>.deriv |>.symm rw [along_congr g d Filter.EventuallyEq.rfl hdtrans] at hchange exact hchange theorem exists_geodesic_local_flow (g : SmoothMetric M) {c : OpenPartialHomeomorph M E3} (hc : c ∈ atlas E3 M) {z : E3 × E3} (hz : z.1 ∈ c.target) : ∃ r > (0 : ℝ), ∃ ε > (0 : ℝ), ∃ X : (E3 × E3) → ℝ → E3 × E3, (∀ y ∈ Metric.ball z r, X y 0 = y) ∧ (∀ y ∈ Metric.ball z r, IsGeodesicOn g (fun t => c.symm (X y t).1) (Ioo (-ε) ε)) ∧ (∀ y ∈ Metric.ball z r, ∀ t ∈ Ioo (-ε) ε, HasDerivAt (X y) (geodesicPhase g c (X y t)) t) ∧ (∀ y ∈ Metric.ball z r, ∀ t ∈ Ioo (-ε) ε, (X y t).1 ∈ c.target) ∧ ContDiffOn ℝ 1 (Function.uncurry X) (Metric.ball z r ×ˢ Ioo (-ε) ε) := by obtain ⟨r,hr,ε,hε,X,h0,hd,hs⟩ := ODEVariation.exists_contDiff_flow (contDiffAt_infty.mp (geodesicPhase_smoothAt g hc hz) 2) have hO : IsOpen (Metric.ball z r ×ˢ Ioo (-ε) ε) := Metric.isOpen_ball.prod isOpen_Ioo have hp : (z,(0:ℝ)) ∈ Metric.ball z r ×ˢ Ioo (-ε) ε := ⟨Metric.mem_ball_self hr,by constructor <;> linarith⟩ have hx : ContinuousAt (Function.uncurry X) (z,0) := hs.continuousOn.continuousAt (hO.mem_nhds hp) have hmem : {p : (E3 × E3) × ℝ | (X p.1 p.2).1 ∈ c.target} ∈ 𝓝 (z,0) := by apply hx.fst.preimage_mem_nhds simpa only [Function.uncurry_apply_pair,h0 z (Metric.mem_ball_self hr)] using c.open_target.mem_nhds hz obtain ⟨η,hη,hsmall⟩ := Metric.mem_nhds_iff.mp (inter_mem hmem (hO.mem_nhds hp)) have hsub : ∀ y ∈ Metric.ball z (η/2), ∀ t ∈ Ioo (-(η/2)) (η/2), (X y t).1 ∈ c.target ∧ (y,t) ∈ Metric.ball z r ×ˢ Ioo (-ε) ε := by intro y hy t ht apply hsmall (a := (y,t)) rw [Metric.mem_ball,Prod.dist_eq,max_lt_iff] exact ⟨lt_trans hy (half_lt_self hη),by rw [Real.dist_eq,sub_zero,abs_lt]; constructor <;> linarith [ht.1,ht.2]⟩ refine ⟨η/2,half_pos hη,η/2,half_pos hη,X,?_,?_,?_,?_,?_⟩ · intro y hy exact h0 y (hsub y hy 0 ⟨by linarith,by linarith⟩).2.1 · intro y hy exact chart_geodesic g hc isOpen_Ioo (fun t ht => (hsub y hy t ht).1) (fun t ht => hd y (hsub y hy t ht).2.1 t (hsub y hy t ht).2.2) · intro y hy t ht exact hd y (hsub y hy t ht).2.1 t (hsub y hy t ht).2.2 · exact fun y hy t ht => (hsub y hy t ht).1 · exact hs.mono (fun p hp => (hsub p.1 hp.1 p.2 hp.2).2) end ThreeManifold.RiemannianCoordinates end section open Set Filter Topology Bundle namespace ThreeManifold theorem fiberBundle_t2 {B F : Type*} {E : B → Type*} [TopologicalSpace B] [T2Space B] [TopologicalSpace F] [T2Space F] [TopologicalSpace (TotalSpace F E)] [∀ x, TopologicalSpace (E x)] [FiberBundle F E] : T2Space (TotalSpace F E) := by classical rw [t2Space_iff_disjoint_nhds] intro p q hpq by_cases hb : p.1 = q.1 · let e := trivializationAt F E p.1 have hp : p ∈ e.source := e.mem_source.mpr (FiberBundle.mem_baseSet_trivializationAt' p.1) have hq : q ∈ e.source := e.mem_source.mpr (hb ▸ FiberBundle.mem_baseSet_trivializationAt' p.1) have he : e p ≠ e q := fun h => hpq (e.toOpenPartialHomeomorph.injOn hp hq h) exact Filter.disjoint_of_map ((disjoint_nhds_nhds.mpr he).mono (e.toOpenPartialHomeomorph.continuousAt hp) (e.toOpenPartialHomeomorph.continuousAt hq)) · exact Filter.disjoint_of_map ((disjoint_nhds_nhds.mpr hb).mono (FiberBundle.continuous_proj F E).continuousAt (FiberBundle.continuous_proj F E).continuousAt) end ThreeManifold end section open Set Filter Topology Bundle Manifold IsManifold open scoped ContDiff namespace ThreeManifold.RiemannianCoordinates variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] def phaseLift (γ : ℝ → M) (t : ℝ) : TangentBundle I3 M := ⟨γ t,velocity γ t⟩ def chartPhase (c : OpenPartialHomeomorph M E3) (γ : ℝ → M) (t : ℝ) : E3 × E3 := (c (γ t),deriv (c ∘ γ) t) lemma phaseLift_chart {γ : ℝ → M} {t : ℝ} (hγ : ContMDiffAt 𝓘(ℝ,ℝ) I3 ∞ γ t) {p : TangentBundle I3 M} (hp : γ t ∈ (chartAt E3 p.1).source) : chartAt (ModelProd E3 E3) p (phaseLift γ t) = chartPhase (chartAt E3 p.1) γ t := by apply Prod.ext · rfl · have he := congrArg (fun q : TangentBundle I3 E3 => q.2) (tangentMap_chart (I := I3) (p := p) (q := phaseLift γ t) hp) exact he.symm.trans (velocity_coordinates hγ (chart_mem_atlas E3 p.1) hp) lemma phaseLift_continuousAt {γ : ℝ → M} {t : ℝ} {U : Set ℝ} (hU : IsOpen U) (hγU : ContMDiffOn 𝓘(ℝ,ℝ) I3 ∞ γ U) (htU : t ∈ U) : ContinuousAt (phaseLift γ) t := by have hγ := hγU.contMDiffAt (hU.mem_nhds htU) let p := phaseLift γ t let c := chartAt E3 (γ t) have ht : γ t ∈ c.source := mem_chart_source E3 (γ t) have hn : γ ⁻¹' c.source ∈ 𝓝 t := hγ.continuousAt.preimage_mem_nhds (c.open_source.mem_nhds ht) apply ((chartAt (ModelProd E3 E3) p).continuousAt_iff_continuousAt_comp_left (show (phaseLift γ) ⁻¹' (chartAt (ModelProd E3 E3) p).source ∈ 𝓝 t by filter_upwards [hn] with r hr exact (TangentBundle.mem_chart_source_iff _ _).mpr hr)).mpr have hcγ : ContDiffAt ℝ ∞ (c ∘ γ) t := contMDiffAt_iff_contDiffAt.mp ((contMDiffAt_of_mem_maximalAtlas (subset_maximalAtlas (chart_mem_atlas E3 (γ t))) ht).comp t hγ) have hs : ∀ᶠ r in 𝓝 t, ContMDiffAt 𝓘(ℝ,ℝ) I3 ∞ γ r := by filter_upwards [hU.mem_nhds htU] with r hr exact hγU.contMDiffAt (hU.mem_nhds hr) apply (hcγ.continuousAt.prodMk (hcγ.derivWithin (m := ∞) (by simp)).continuousAt).congr_of_eventuallyEq filter_upwards [hn,hs] with r hr hrs exact phaseLift_chart hrs hr lemma chartPhase_contDiffAt {γ : ℝ → M} {t : ℝ} (hγ : ContMDiffAt 𝓘(ℝ,ℝ) I3 ∞ γ t) {c : OpenPartialHomeomorph M E3} (hc : c ∈ atlas E3 M) (ht : γ t ∈ c.source) : ContDiffAt ℝ ∞ (chartPhase c γ) t := by have h : ContDiffAt ℝ ∞ (c ∘ γ) t := contMDiffAt_iff_contDiffAt.mp ((contMDiffAt_of_mem_maximalAtlas (subset_maximalAtlas hc) ht).comp t hγ) exact h.prodMk (h.derivWithin (m := ∞) (by simp)) lemma geodesic_chartPhase_ode (g : SmoothMetric M) {γ : ℝ → M} {U : Set ℝ} (hU : IsOpen U) (hγ : IsGeodesicOn g γ U) {t : ℝ} (ht : t ∈ U) {c : OpenPartialHomeomorph M E3} (hc : c ∈ atlas E3 M) (hct : γ t ∈ c.source) : HasDerivAt (chartPhase c γ) (geodesicPhase g c (chartPhase c γ t)) t := by have hs := chartPhase_contDiffAt (hγ.1.contMDiffAt (hU.mem_nhds ht)) hc hct have hd := hs.snd.differentiableAt (by simp) |>.hasDerivAt have he := hγ.2 c hc t ht hct change deriv (deriv (c ∘ γ)) t + connectionTerm g c (c (γ t)) (deriv (c ∘ γ) t) (deriv (c ∘ γ) t) = 0 at he have he' : deriv (deriv (c ∘ γ)) t = -connectionTerm g c (c (γ t)) (deriv (c ∘ γ) t) (deriv (c ∘ γ) t) := eq_neg_of_add_eq_zero_left he exact (hs.fst.differentiableAt (by simp)).hasDerivAt.prodMk (hd.congr_deriv he') lemma geodesic_eventuallyEq (g : SmoothMetric M) {γ η : ℝ → M} {U : Set ℝ} (hU : IsOpen U) (hγ : IsGeodesicOn g γ U) (hη : IsGeodesicOn g η U) {t : ℝ} (ht : t ∈ U) (he : phaseLift γ t = phaseLift η t) : γ =ᶠ[𝓝 t] η := by let c := chartAt E3 (γ t) have hc := chart_mem_atlas E3 (γ t) have hct : γ t ∈ c.source := mem_chart_source E3 (γ t) have hbase : γ t = η t := congrArg Bundle.TotalSpace.proj he have hηt : η t ∈ c.source := hbase ▸ hct have hγs := hγ.1.contMDiffAt (hU.mem_nhds ht) have hηs := hη.1.contMDiffAt (hU.mem_nhds ht) have hinitial : chartPhase c γ t = chartPhase c η t := by have h1 := phaseLift_chart (p := phaseLift γ t) hγs hct have h2 := phaseLift_chart (p := phaseLift γ t) hηs hηt change _ = chartPhase c γ t at h1 change _ = chartPhase c η t at h2 rw [← h1, ← h2, he] obtain ⟨K,S,hS,hL⟩ := (geodesicPhase_smoothAt g (z := chartPhase c γ t) hc (c.map_source hct)).of_le (show (1:ℕ∞ω) ≤ ∞ by simp) |>.exists_lipschitzOnWith have hγn : ∀ᶠ r in 𝓝 t, r ∈ U ∧ γ r ∈ c.source ∧ chartPhase c γ r ∈ S := Filter.Eventually.and (hU.mem_nhds ht) (Filter.Eventually.and (hγs.continuousAt.preimage_mem_nhds (c.open_source.mem_nhds hct)) ((chartPhase_contDiffAt hγs hc hct).continuousAt.preimage_mem_nhds hS)) have hηn : ∀ᶠ r in 𝓝 t, r ∈ U ∧ η r ∈ c.source ∧ chartPhase c η r ∈ S := Filter.Eventually.and (hU.mem_nhds ht) (Filter.Eventually.and (hηs.continuousAt.preimage_mem_nhds (c.open_source.mem_nhds hηt)) ((chartPhase_contDiffAt hηs hc hηt).continuousAt.preimage_mem_nhds (hinitial ▸ hS))) have hode := ODE_solution_unique_of_eventually (v := fun _ => geodesicPhase g c) (s := fun _ => S) (Filter.Eventually.of_forall (fun _ => hL)) (hγn.mono (fun r hr => ⟨geodesic_chartPhase_ode g hU hγ hr.1 hc hr.2.1,hr.2.2⟩)) (hηn.mono (fun r hr => ⟨geodesic_chartPhase_ode g hU hη hr.1 hc hr.2.1,hr.2.2⟩)) hinitial filter_upwards [hode,hγn,hηn] with r hr hgr her exact c.injOn hgr.2.1 her.2.1 (congrArg Prod.fst hr) lemma phaseLift_eventuallyEq {γ η : ℝ → M} {t : ℝ} (he : γ =ᶠ[𝓝 t] η) : phaseLift γ =ᶠ[𝓝 t] phaseLift η := by let _ : IsManifold I3 ∞ M := inferInstance filter_upwards [he.eventually_nhds] with r hr change γ =ᶠ[𝓝 r] η at hr have hb := hr.eq_of_nhds have hd := congrArg (fun L : ℝ →L[ℝ] E3 => L 1) (hr.mfderiv_eq (I := 𝓘(ℝ,ℝ)) (I' := I3)) unfold phaseLift velocity exact Bundle.TotalSpace.ext hb (heq_of_eq hd) theorem geodesic_unique [T2Space M] (g : SmoothMetric M) {γ η : ℝ → M} {U : Set ℝ} (hU : IsOpen U) (hUc : OrdConnected U) (hγ : IsGeodesicOn g γ U) (hη : IsGeodesicOn g η U) {t : ℝ} (ht : t ∈ U) (he : phaseLift γ t = phaseLift η t) : EqOn γ η U := by let : T2Space (TangentBundle I3 M) := fiberBundle_t2 let Z : Set U := {s | phaseLift γ s = phaseLift η s} let : PreconnectedSpace U := Subtype.preconnectedSpace hUc.isPreconnected have hZ : IsClopen Z := by constructor · apply isClosed_eq · exact continuous_iff_continuousAt.mpr (fun s => (phaseLift_continuousAt hU hγ.1 s.property).comp continuous_subtype_val.continuousAt) · exact continuous_iff_continuousAt.mpr (fun s => (phaseLift_continuousAt hU hη.1 s.property).comp continuous_subtype_val.continuousAt) · apply isOpen_iff_mem_nhds.mpr intro s hs exact continuous_subtype_val.continuousAt.preimage_mem_nhds (phaseLift_eventuallyEq (geodesic_eventuallyEq g hU hγ hη s.property hs)) have hZu := hZ.eq_univ ⟨⟨t,ht⟩,he⟩ intro s hs have hsz : (⟨s,hs⟩ : U) ∈ Z := by rw [hZu]; trivial exact congrArg Bundle.TotalSpace.proj hsz end ThreeManifold.RiemannianCoordinates end section open Set Filter Topology Bundle namespace ThreeManifold.CompactEnergy variable {B : Type*} [TopologicalSpace B] [CompactSpace B] [T2Space B] variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [ProperSpace F] variable {E : B → Type*} [TopologicalSpace (TotalSpace F E)] [∀ x, NormedAddCommGroup (E x)] [∀ x, InnerProductSpace ℝ (E x)] [FiberBundle F E] [VectorBundle ℝ F E] [IsContinuousRiemannianBundle F E] lemma continuous_fiberNorm : Continuous (fun p : TotalSpace F E => ‖p.2‖) := by let _ : CompactSpace B := inferInstance let _ : T2Space B := inferInstance let _ : ProperSpace F := inferInstance have h : Continuous (fun p : TotalSpace F E => inner ℝ p.2 p.2) := Continuous.inner_bundle continuous_id continuous_id convert h.sqrt using 1 ext p simp theorem isCompact_fiberDisk (R : ℝ) : IsCompact {p : TotalSpace F E | ‖p.2‖ ≤ R} := by classical have hlocal : ∀ x : B, ∃ C > (0:ℝ), ∃ K : Set B, K ∈ 𝓝 x ∧ IsCompact K ∧ K ⊆ (trivializationAt F E x).baseSet ∧ ∀ y ∈ K, ‖(trivializationAt F E x).continuousLinearMapAt ℝ y‖ ≤ C := by intro x obtain ⟨C,hC,hb⟩ := eventually_norm_trivializationAt_lt F E x obtain ⟨K,hK,hsub,hcomp⟩ := local_compact_nhds (inter_mem hb ((trivializationAt F E x).open_baseSet.mem_nhds (FiberBundle.mem_baseSet_trivializationAt' x))) exact ⟨C,hC,K,hK,hcomp,fun y hy => (hsub hy).2,fun y hy => (hsub hy).1.le⟩ choose C hC K hKn hKc hKb hbound using hlocal obtain ⟨T,_,hcover⟩ := isCompact_univ.elim_nhds_subcover K (fun x _ => hKn x) let L : B → Set (TotalSpace F E) := fun x => (trivializationAt F E x).toOpenPartialHomeomorph.symm '' (K x ×ˢ Metric.closedBall (0:F) (C x * max R 0)) have hLc : ∀ x, IsCompact (L x) := by intro x apply ((hKc x).prod (isCompact_closedBall _ _)).image_of_continuousOn exact (trivializationAt F E x).toOpenPartialHomeomorph.continuousOn_symm.mono (fun z hz => (trivializationAt F E x).mem_target.mpr (hKb x hz.1)) apply (T.isCompact_biUnion (fun x _ => hLc x)).of_isClosed_subset (isClosed_le continuous_fiberNorm continuous_const) intro p hp obtain ⟨x,hx,hpx⟩ := mem_iUnion₂.mp (hcover (mem_univ p.1)) apply mem_iUnion₂.mpr refine ⟨x,hx,?_⟩ let e := trivializationAt F E x have hps : p ∈ e.source := e.mem_source.mpr (hKb x hpx) refine ⟨e p,⟨?_,?_⟩,e.symm_apply_apply hps⟩ · simpa only [e.coe_fst hps] using hpx · rw [Metric.mem_closedBall,dist_zero_right] have he : (e p).2 = e.continuousLinearMapAt ℝ p.1 p.2 := by exact (e.continuousLinearMapAt_apply_of_mem ℝ (hKb x hpx) p.2).symm rw [he] calc ‖e.continuousLinearMapAt ℝ p.1 p.2‖ ≤ ‖e.continuousLinearMapAt ℝ p.1‖ * ‖p.2‖ := ContinuousLinearMap.le_opNorm _ _ _ ≤ C x * max R 0 := mul_le_mul (hbound x p.1 hpx) (hp.trans (le_max_left _ _)) (norm_nonneg _) (hC x).le end ThreeManifold.CompactEnergy end section open Set Filter Topology Bundle Manifold open scoped ContDiff namespace ThreeManifold.RiemannianCoordinates variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] [CompactSpace M] [T2Space M] def phaseEnergy (g : SmoothMetric M) (p : TangentBundle I3 M) : ℝ := g.inner p.1 p.2 p.2 lemma continuous_phaseEnergy (g : SmoothMetric M) : Continuous (phaseEnergy g) := by let _ : CompactSpace M := inferInstance let _ : T2Space M := inferInstance let : RiemannianBundle (fun x : M => TangentSpace I3 x) := ⟨g.toRiemannianMetric⟩ let : IsContinuousRiemannianBundle E3 (fun x : M => TangentSpace I3 x) := ⟨g.inner, g.contMDiff.continuous, fun _ _ _ => rfl⟩ have h : Continuous (fun p : TangentBundle I3 M => inner ℝ p.2 p.2) := Continuous.inner_bundle continuous_id continuous_id exact h theorem isCompact_phaseEnergy (g : SmoothMetric M) (R : ℝ) : IsCompact {p : TangentBundle I3 M | phaseEnergy g p ≤ R} := by let : RiemannianBundle (fun x : M => TangentSpace I3 x) := ⟨g.toRiemannianMetric⟩ let : IsContinuousRiemannianBundle E3 (fun x : M => TangentSpace I3 x) := ⟨g.inner, g.contMDiff.continuous, fun _ _ _ => rfl⟩ apply (CompactEnergy.isCompact_fiberDisk (F := E3) (E := fun x : M => TangentSpace I3 x) (Real.sqrt (max R 0))).of_isClosed_subset (isClosed_le (continuous_phaseEnergy g) continuous_const) intro p hp change ‖p.2‖ ≤ Real.sqrt (max R 0) rw [norm_eq_sqrt_real_inner] apply Real.sqrt_le_sqrt exact hp.trans (le_max_left _ _) end ThreeManifold.RiemannianCoordinates end section open Set Filter Topology namespace ThreeManifold.FlowContinuation structure Equation (X : Type*) where Holds : (ℝ → X) → ℝ → Prop germ {z w : ℝ → X} {t : ℝ} : z =ᶠ[𝓝 t] w → Holds z t → Holds w t shift (z : ℝ → X) (a t : ℝ) : Holds (fun r => z (r+a)) t ↔ Holds z (t+a) unique {z w : ℝ → X} {a b t : ℝ} : (∀ r ∈ Ioo a b, Holds z r) → (∀ r ∈ Ioo a b, Holds w r) → t ∈ Ioo a b → z t = w t → EqOn z w (Ioo a b) namespace Equation variable {X : Type*} (P : Equation X) def CurveOn (z : ℝ → X) (U : Set ℝ) : Prop := ∀ t ∈ U, P.Holds z t lemma CurveOn.mono {z : ℝ → X} {U V : Set ℝ} (h : P.CurveOn z U) (hVU : V ⊆ U) : P.CurveOn z V := fun t ht => h t (hVU ht) lemma CurveOn.congr {z w : ℝ → X} {U : Set ℝ} (hU : IsOpen U) (h : P.CurveOn z U) (he : EqOn z w U) : P.CurveOn w U := by intro t ht exact P.germ (Filter.eventually_of_mem (hU.mem_nhds ht) (fun r hr => he hr)) (h t ht) lemma CurveOn.comp_add {z : ℝ → X} {U : Set ℝ} (h : P.CurveOn z U) (a : ℝ) : P.CurveOn (fun r => z (r+a)) ((fun r => r+a) ⁻¹' U) := by intro t ht exact (P.shift z a t).mpr (h (t+a) ht) lemma eqOn_piecewise {z w : ℝ → X} {a b c d t : ℝ} (hz : P.CurveOn z (Ioo a b)) (hw : P.CurveOn w (Ioo c d)) (ht : t ∈ Ioo a b ∩ Ioo c d) (he : z t = w t) : EqOn (piecewise (Ioo a b) z w) w (Ioo c d) := by classical have H : EqOn z w (Ioo (max a c) (min b d)) := P.unique (hz.mono P (Ioo_subset_Ioo (le_max_left ..) (min_le_left ..))) (hw.mono P (Ioo_subset_Ioo (le_max_right ..) (min_le_right ..))) ⟨max_lt ht.1.1 ht.2.1,lt_min ht.1.2 ht.2.2⟩ he intro r hr by_cases hrz : r ∈ Ioo a b · rw [piecewise,ite_eq_left hrz] exact H ⟨max_lt hrz.1 hr.1,lt_min hrz.2 hr.2⟩ · rw [piecewise,ite_eq_right hrz] lemma curveOn_piecewise {z w : ℝ → X} {a b c d t : ℝ} (hz : P.CurveOn z (Ioo a b)) (hw : P.CurveOn w (Ioo c d)) (ht : t ∈ Ioo a b ∩ Ioo c d) (he : z t = w t) : P.CurveOn (piecewise (Ioo a b) z w) (Ioo a b ∪ Ioo c d) := by classical intro r hr by_cases hrz : r ∈ Ioo a b · exact P.germ (Filter.eventually_of_mem (isOpen_Ioo.mem_nhds hrz) (fun s hs => (piecewise_eq_of_mem (Ioo a b) z w hs).symm)) (hz r hrz) · have hrw := hr.resolve_left hrz exact P.germ (Filter.eventually_of_mem (isOpen_Ioo.mem_nhds hrw) (fun s hs => (P.eqOn_piecewise hz hw ht he hs).symm)) (hw r hrw) lemma global_of_arbitrarily_long {x : X} (h : ∀ a : ℝ, ∃ z, z 0 = x ∧ P.CurveOn z (Ioo (-a) a)) : ∃ z, z 0 = x ∧ ∀ t, P.Holds z t := by choose z h0 hz using h have he {a b : ℝ} (ha : 0 < a) (hab : a ≤ b) : EqOn (z a) (z b) (Ioo (-a) a) := P.unique (hz a) ((hz b).mono P (Ioo_subset_Ioo (neg_le_neg hab) hab)) ⟨neg_lt_zero.mpr ha,ha⟩ ((h0 a).trans (h0 b).symm) let w : ℝ → X := fun t => z (|t|+1) t have hw {a : ℝ} : EqOn w (z a) (Ioo (-a) a) := by intro t ht by_cases hta : |t|+1 ≤ a · exact he (by positivity) hta (abs_lt.mp (lt_add_one _)) · exact (he (by linarith [ht.1,ht.2]) (le_of_not_ge hta) ht).symm refine ⟨w,h0 (|0|+1),fun t => ?_⟩ have ht : t ∈ Ioo (-(|t|+1)) (|t|+1) := abs_lt.mp (lt_add_one _) exact P.germ (Filter.eventually_of_mem (isOpen_Ioo.mem_nhds ht) (fun r hr => (hw hr).symm)) (hz (|t|+1) t ht) theorem exists_global_of_uniform_invariant {K : Set X} {ε : ℝ} (hε : 0 < ε) (hloc : ∀ x ∈ K, ∃ z : ℝ → X, z 0 = x ∧ P.CurveOn z (Ioo (-ε) ε)) (hinv : ∀ (z : ℝ → X) (a : ℝ), 0 < a → z 0 ∈ K → P.CurveOn z (Ioo (-a) a) → MapsTo z (Ioo (-a) a) K) {x : X} (hx : x ∈ K) : ∃ z, z 0 = x ∧ ∀ t, P.Holds z t := by classical let S := {a : ℝ | ∃ z, z 0 = x ∧ P.CurveOn z (Ioo (-a) a)} have hεS : ε ∈ S := hloc x hx apply P.global_of_arbitrarily_long suffices hnb : ¬ BddAbove S by intro a obtain ⟨b,hb,hab⟩ := not_bddAbove_iff.mp hnb a obtain ⟨z,hz0,hz⟩ := hb exact ⟨z,hz0,hz.mono P (Ioo_subset_Ioo (neg_le_neg hab.le) hab.le)⟩ intro hbdd let A := sSup S have hεA : ε ≤ A := le_csSup hbdd hεS obtain ⟨a,⟨z,hz0,hz⟩,ha⟩ := Real.add_neg_lt_sSup (⟨ε,hεS⟩ : S.Nonempty) (ε := -(ε/2)) (by linarith) change A + -(ε/2) < a at ha have ha0 : 0 < a := by linarith have hleft : -(A-ε/2) ∈ Ioo (-a) a := ⟨by linarith,by linarith⟩ have hright : A-ε/2 ∈ Ioo (-a) a := ⟨by linarith,by linarith⟩ have hzK := hinv z a ha0 (hz0 ▸ hx) hz obtain ⟨z1,h10,h1⟩ := hloc (z (-(A-ε/2))) (hzK hleft) obtain ⟨z2,h20,h2⟩ := hloc (z (A-ε/2)) (hzK hright) let w1 : ℝ → X := fun t => z1 (t+(A-ε/2)) let w2 : ℝ → X := fun t => z2 (t-(A-ε/2)) have hw1 : P.CurveOn w1 (Ioo (-(A+ε/2)) (-(A-3*ε/2))) := by apply (h1.comp_add P (A-ε/2)).mono P intro t ht; change -ε < t+(A-ε/2) ∧ t+(A-ε/2) < ε constructor <;> linarith [ht.1,ht.2] have hw2 : P.CurveOn w2 (Ioo (A-3*ε/2) (A+ε/2)) := by apply (h2.comp_add P (-(A-ε/2))).mono P intro t ht; change -ε < t+ -(A-ε/2) ∧ t+ -(A-ε/2) < ε constructor <;> linarith [ht.1,ht.2] have he1 : z (-(A-ε/2)) = w1 (-(A-ε/2)) := by simp [w1,h10] have he2 : z (A-ε/2) = w2 (A-ε/2) := by simp [w2,h20] let w : ℝ → X := piecewise (Ioo (-(A+ε/2)) a) (piecewise (Ioo (-a) a) z w1) w2 have hw0 : w 0 = x := by simp only [w,piecewise,ite_eq_left (show 0 ∈ Ioo (-(A+ε/2)) a from ⟨by linarith,ha0⟩), ite_eq_left (show 0 ∈ Ioo (-a) a from ⟨by linarith,ha0⟩),hz0] have hw : P.CurveOn w (Ioo (-(A+ε/2)) (A+ε/2)) := by apply (P.curveOn_piecewise (t := A-ε/2) ?_ hw2 ⟨⟨by linarith,by linarith⟩,⟨by linarith,by linarith⟩⟩ (by rw [piecewise,ite_eq_left hright]; exact he2)).mono P (Ioo_subset_Ioo_union_Ioo le_rfl (by linarith) (by linarith)) exact (P.curveOn_piecewise hz hw1 ⟨hleft,⟨by linarith,by linarith⟩⟩ he1).mono P (union_comm _ _ ▸ Ioo_subset_Ioo_union_Ioo (by linarith) (by linarith) le_rfl) exact (not_lt.mpr (le_csSup hbdd (show A+ε/2 ∈ S from ⟨w,hw0,hw⟩))) (lt_add_of_pos_right A (half_pos hε)) theorem exists_global_of_compact_invariant [TopologicalSpace X] {K : Set X} (hK : IsCompact K) (hloc : ∀ x ∈ K, ∃ U ∈ 𝓝 x, ∃ ε > (0:ℝ), ∀ y ∈ U, ∃ z, z 0 = y ∧ P.CurveOn z (Ioo (-ε) ε)) (hinv : ∀ (z : ℝ → X) (a : ℝ), 0 < a → z 0 ∈ K → P.CurveOn z (Ioo (-a) a) → MapsTo z (Ioo (-a) a) K) {x : X} (hx : x ∈ K) : ∃ z, z 0 = x ∧ ∀ t, P.Holds z t := by classical choose U hUn ε hε hsol using hloc obtain ⟨T,hcover⟩ := hK.elim_nhds_subcover' U hUn have hT : T.Nonempty := by obtain ⟨i,hi,_⟩ := mem_iUnion₂.mp (hcover hx) exact ⟨i,hi⟩ let e := T.inf' hT (fun i => ε i i.property) have he : 0 < e := (Finset.lt_inf'_iff hT).mpr (fun i _ => hε i i.property) apply P.exists_global_of_uniform_invariant he ?_ hinv hx intro y hy obtain ⟨i,hi,hyi⟩ := mem_iUnion₂.mp (hcover hy) obtain ⟨z,hz0,hz⟩ := hsol i i.property y hyi have hei : e ≤ ε i i.property := Finset.inf'_le _ hi exact ⟨z,hz0,hz.mono P (Ioo_subset_Ioo (neg_le_neg hei) hei)⟩ end Equation end ThreeManifold.FlowContinuation end section open Set Filter Topology Bundle Manifold IsManifold open scoped ContDiff namespace ThreeManifold.RiemannianCoordinates variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] lemma deriv_comp_translate {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a t : ℝ} (hf : DifferentiableAt ℝ f (t+a)) : deriv (fun r => f (r+a)) t = deriv f (t+a) := by exact (hf.hasFDerivAt.comp_hasDerivAt t ((hasDerivAt_id t).add_const a)).deriv lemma geodesic_translate (g : SmoothMetric M) {γ : ℝ → M} {U : Set ℝ} (hU : IsOpen U) (hγ : IsGeodesicOn g γ U) (a : ℝ) : IsGeodesicOn g (fun r => γ (r+a)) ((fun r => r+a) ⁻¹' U) := by have htA : ContMDiff 𝓘(ℝ,ℝ) 𝓘(ℝ,ℝ) ∞ (fun r : ℝ => r+a) := contMDiff_iff_contDiff.mpr (contDiff_id.add contDiff_const) refine ⟨hγ.1.comp htA.contMDiffOn (fun _ h => h),?_⟩ intro c hc t ht hct have hs := chartPhase_contDiffAt (hγ.1.contMDiffAt (hU.mem_nhds ht)) hc hct have hn : ∀ᶠ r in 𝓝 t, r+a ∈ U ∧ γ (r+a) ∈ c.source := continuousAt_id.add continuousAt_const |>.preimage_mem_nhds (Filter.Eventually.and (hU.mem_nhds ht) ((hγ.1.contMDiffAt (hU.mem_nhds ht)).continuousAt.preimage_mem_nhds (c.open_source.mem_nhds hct))) have hd : deriv (c ∘ (fun r => γ (r+a))) =ᶠ[𝓝 t] fun r => deriv (c ∘ γ) (r+a) := by filter_upwards [hn] with r hr exact deriv_comp_translate ((chartPhase_contDiffAt (hγ.1.contMDiffAt (hU.mem_nhds hr.1)) hc hr.2).fst.differentiableAt (by simp)) have hdd : deriv (deriv (c ∘ (fun r => γ (r+a)))) t = deriv (deriv (c ∘ γ)) (t+a) := hd.deriv_eq.trans (deriv_comp_translate (hs.snd.differentiableAt (by simp))) change _ + _ = 0 rw [hdd,hd.eq_of_nhds] exact hγ.2 c hc (t+a) ht hct lemma phaseLift_translate {γ : ℝ → M} {a t : ℝ} (hγ : ContMDiffAt 𝓘(ℝ,ℝ) I3 ∞ γ (t+a)) : phaseLift (fun r => γ (r+a)) t = phaseLift γ (t+a) := by let p := phaseLift γ (t+a) let c := chartAt (ModelProd E3 E3) p have htrans : ContMDiffAt 𝓘(ℝ,ℝ) I3 ∞ (fun r => γ (r+a)) t := hγ.comp t (contMDiffAt_iff_contDiffAt.mpr (contDiffAt_id.add contDiffAt_const)) have hs : phaseLift (fun r => γ (r+a)) t ∈ c.source := (TangentBundle.mem_chart_source_iff _ _).mpr (mem_chart_source E3 (γ (t+a))) apply c.injOn hs (mem_chart_source (ModelProd E3 E3) p) rw [phaseLift_chart htrans (mem_chart_source E3 (γ (t+a))), phaseLift_chart hγ (mem_chart_source E3 (γ (t+a)))] apply Prod.ext · rfl exact deriv_comp_translate ((chartPhase_contDiffAt hγ (chart_mem_atlas E3 (γ (t+a))) (mem_chart_source E3 (γ (t+a)))).fst.differentiableAt (by simp)) def PhaseGeodesicAt (g : SmoothMetric M) (z : ℝ → TangentBundle I3 M) (t : ℝ) : Prop := ∃ (γ : ℝ → M) (U : Set ℝ), IsOpen U ∧ t ∈ U ∧ IsGeodesicOn g γ U ∧ EqOn z (phaseLift γ) U lemma phaseGeodesic_germ (g : SmoothMetric M) {z w : ℝ → TangentBundle I3 M} {t : ℝ} (he : z =ᶠ[𝓝 t] w) (hz : PhaseGeodesicAt g z t) : PhaseGeodesicAt g w t := by obtain ⟨γ,U,hU,ht,hγ,hz⟩ := hz obtain ⟨V,hV,hVo,htV⟩ := mem_nhds_iff.mp he exact ⟨γ,U ∩ V,hU.inter hVo,⟨ht,htV⟩,⟨hγ.1.mono inter_subset_left, fun c hc s hs hsc => hγ.2 c hc s hs.1 hsc⟩, fun s hs => (hV hs.2).symm.trans (hz hs.1)⟩ lemma phaseGeodesic_translate (g : SmoothMetric M) {z : ℝ → TangentBundle I3 M} {a t : ℝ} (hz : PhaseGeodesicAt g z (t+a)) : PhaseGeodesicAt g (fun r => z (r+a)) t := by obtain ⟨γ,U,hU,ht,hγ,hz⟩ := hz refine ⟨fun r => γ (r+a),(fun r => r+a) ⁻¹' U, hU.preimage (continuous_id.add continuous_const),ht,geodesic_translate g hU hγ a,?_⟩ intro r hr exact (hz hr).trans (phaseLift_translate (hγ.1.contMDiffAt (hU.mem_nhds hr))).symm lemma phaseGeodesicOn_base (g : SmoothMetric M) {z : ℝ → TangentBundle I3 M} {U : Set ℝ} (_hU : IsOpen U) (hz : ∀ t ∈ U, PhaseGeodesicAt g z t) : IsGeodesicOn g (fun t => (z t).proj) U ∧ EqOn z (phaseLift (fun t => (z t).proj)) U := by have hlocal (t : ℝ) (ht : t ∈ U) : ∃ γ V, IsOpen V ∧ t ∈ V ∧ IsGeodesicOn g γ V ∧ (fun r => (z r).proj) =ᶠ[𝓝 t] γ ∧ z t = phaseLift γ t := by obtain ⟨γ,V,hV,htV,hγ,he⟩ := hz t ht exact ⟨γ,V,hV,htV,hγ,(Filter.eventually_of_mem (hV.mem_nhds htV) (fun r hr => congrArg Bundle.TotalSpace.proj (he hr))),he htV⟩ refine ⟨⟨?_,?_⟩,?_⟩ · intro t ht obtain ⟨γ,V,hV,htV,hγ,he,_⟩ := hlocal t ht exact ((hγ.1.contMDiffAt (hV.mem_nhds htV)).congr_of_eventuallyEq he).contMDiffWithinAt · intro c hc t ht hct obtain ⟨γ,V,hV,htV,hγ,he,_⟩ := hlocal t ht have hec := he.fun_comp c rw [along_congr g c hec hec.deriv] exact hγ.2 c hc t htV (he.eq_of_nhds ▸ hct) · intro t ht obtain ⟨γ,V,hV,htV,hγ,he,het⟩ := hlocal t ht exact het.trans (phaseLift_eventuallyEq he).eq_of_nhds.symm def geodesicEquation [T2Space M] (g : SmoothMetric M) : FlowContinuation.Equation (TangentBundle I3 M) where Holds := PhaseGeodesicAt g germ := phaseGeodesic_germ g shift z a t := by constructor · intro h have hh := phaseGeodesic_translate g (a := -a) (t := t+a) (by simpa using h) simpa only [add_assoc,neg_add_cancel,add_zero] using hh · exact phaseGeodesic_translate g unique := by intro z w a b t hz hw ht he obtain ⟨hγ,hzp⟩ := phaseGeodesicOn_base g isOpen_Ioo hz obtain ⟨hη,hwp⟩ := phaseGeodesicOn_base g isOpen_Ioo hw have hbase := geodesic_unique g isOpen_Ioo ordConnected_Ioo hγ hη ht ((hzp ht).symm.trans (he.trans (hwp ht))) intro r hr have hphase := phaseLift_eventuallyEq (Filter.eventually_of_mem (isOpen_Ioo.mem_nhds hr) (fun s hs => hbase hs)) exact (hzp hr).trans (hphase.eq_of_nhds.trans (hwp hr).symm) lemma phaseGeodesic_local_existence (g : SmoothMetric M) (p : TangentBundle I3 M) : ∃ U ∈ 𝓝 p, ∃ ε > (0:ℝ), ∀ y ∈ U, ∃ z, z 0 = y ∧ ∀ t ∈ Ioo (-ε) ε, PhaseGeodesicAt g z t := by let c := chartAt E3 p.1 let C := chartAt (ModelProd E3 E3) p have hp : p ∈ C.source := mem_chart_source _ p have hp' : (C p).1 ∈ c.target := c.map_source (mem_chart_source E3 p.1) obtain ⟨r,hr,ε,hε,X,h0,hγ,hd,hXt,_⟩ := exists_geodesic_local_flow g (chart_mem_atlas E3 p.1) hp' let q : TangentBundle I3 M → E3 × E3 := fun y => C y let U := C.source ∩ q ⁻¹' Metric.ball (q p) r have hUn : U ∈ 𝓝 p := inter_mem (C.open_source.mem_nhds hp) ((show ContinuousAt q p from C.continuousAt hp).preimage_mem_nhds (Metric.isOpen_ball.mem_nhds (Metric.mem_ball_self hr))) refine ⟨U,hUn,ε,hε,?_⟩ intro y hy let γ : ℝ → M := fun t => c.symm (X (C y) t).1 have hyb : C y ∈ Metric.ball (q p) r := hy.2 have hzero : (0:ℝ) ∈ Ioo (-ε) ε := ⟨by linarith,by linarith⟩ have hγs : IsGeodesicOn g γ (Ioo (-ε) ε) := hγ (C y) hyb have hm : γ 0 ∈ c.source := c.map_target (hXt (C y) hyb 0 hzero) have hchart : chartPhase c γ 0 = C y := by have he : c ∘ γ =ᶠ[𝓝 (0:ℝ)] fun t => (X (C y) t).1 := Filter.eventually_of_mem (isOpen_Ioo.mem_nhds hzero) (fun t ht => c.right_inv (hXt (C y) hyb t ht)) apply Prod.ext · exact he.eq_of_nhds.trans (congrArg Prod.fst (h0 (C y) hyb)) · have hd1 : HasDerivAt (fun t => (X (C y) t).1) (X (C y) 0).2 0 := (hd (C y) hyb 0 hzero).fst exact he.deriv_eq.trans (hd1.deriv.trans (congrArg Prod.snd (h0 (C y) hyb))) refine ⟨phaseLift γ,?_,fun t ht => ⟨γ,Ioo (-ε) ε,isOpen_Ioo,ht,hγs,fun _ _ => rfl⟩⟩ apply C.injOn ((TangentBundle.mem_chart_source_iff _ _).mpr hm) hy.1 exact (phaseLift_chart (p := p) (hγs.1.contMDiffAt (isOpen_Ioo.mem_nhds hzero)) hm).trans hchart lemma phaseEnergy_conserved [CompactSpace M] [T2Space M] (g : SmoothMetric M) {z : ℝ → TangentBundle I3 M} {U : Set ℝ} (hU : IsOpen U) (hUc : OrdConnected U) (hz : ∀ t ∈ U, PhaseGeodesicAt g z t) {s t : ℝ} (hs : s ∈ U) (ht : t ∈ U) : phaseEnergy g (z s) = phaseEnergy g (z t) := by obtain ⟨hγ,he⟩ := phaseGeodesicOn_base g hU hz let f : ℝ → ℝ := fun r => g.inner (z r).proj (velocity (fun u => (z u).proj) r) (velocity (fun u => (z u).proj) r) have hd : ∀ r ∈ U, HasDerivAt f 0 r := fun r hr => geodesic_speed_hasDerivAt g hU hγ hr have hc : f s = f t := hU.is_const_of_deriv_eq_zero hUc.isPreconnected (fun r hr => (hd r hr).differentiableAt.differentiableWithinAt) (fun r hr => (hd r hr).deriv) hs ht change _ = _ rw [he hs,he ht] exact hc theorem exists_global_geodesic [CompactSpace M] [T2Space M] (g : SmoothMetric M) (p : TangentBundle I3 M) : ∃ γ : ℝ → M, phaseLift γ 0 = p ∧ IsGeodesicOn g γ univ := by let K := {q : TangentBundle I3 M | phaseEnergy g q ≤ phaseEnergy g p} have hinv : ∀ (z : ℝ → TangentBundle I3 M) (a : ℝ), 0 < a → z 0 ∈ K → (geodesicEquation g).CurveOn z (Ioo (-a) a) → MapsTo z (Ioo (-a) a) K := by intro z a ha hzK hcurve t ht change phaseEnergy g (z t) ≤ phaseEnergy g p rw [phaseEnergy_conserved g isOpen_Ioo ordConnected_Ioo hcurve ht (show (0:ℝ) ∈ Ioo (-a) a from ⟨by linarith,ha⟩)] exact hzK obtain ⟨z,hz0,hz⟩ := (geodesicEquation g).exists_global_of_compact_invariant (K := K) (isCompact_phaseEnergy g (phaseEnergy g p)) (fun q _ => phaseGeodesic_local_existence g q) hinv (x := p) (show p ∈ K from (show phaseEnergy g p ≤ phaseEnergy g p from le_rfl)) obtain ⟨hγ,he⟩ := phaseGeodesicOn_base g isOpen_univ (fun t _ => hz t) exact ⟨fun t => (z t).proj,(he (mem_univ 0)).symm.trans hz0,hγ⟩ end ThreeManifold.RiemannianCoordinates end section open Set Filter Topology Manifold open scoped ContDiff namespace ThreeManifold.FlowContinuation variable {E H X : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [TopologicalSpace X] [ChartedSpace H X] [IsManifold I 1 X] lemma flow_contMDiff (F : X → ℝ → X) (h0 : ∀ x, F x 0 = x) (hadd : ∀ x s t, F (F x s) t = F x (s+t)) (hc : ∀ x, Continuous (F x)) (hloc : ∀ x, ContMDiffAt (I.prod 𝓘(ℝ,ℝ)) I 1 (Function.uncurry F) (x,0)) : ContMDiff (I.prod 𝓘(ℝ,ℝ)) I 1 (Function.uncurry F) := by have htime (x : X) (t : ℝ) : ContMDiffAt I I 1 (fun y => F y t) x := by let Z := {s : ℝ | ContMDiffAt I I 1 (fun y => F y s) x} have hstep {s t : ℝ} (hs : s ∈ Z) (hl : ContMDiffAt (I.prod 𝓘(ℝ,ℝ)) I 1 (Function.uncurry F) (F x s,t-s)) : t ∈ Z := by have hh := hl.comp x (hs.prodMk (contMDiffAt_const (c := t-s))) apply hh.congr_of_eventuallyEq exact Filter.Eventually.of_forall (fun y => by change F y t = F (F y s) (t-s) rw [hadd,add_sub_cancel]) have hZc : IsClosed Z := by apply isClosed_of_closure_subset intro t ht have hh := (contMDiffAt_iff_contMDiffAt_nhds (by simp : (1:ℕ∞ω) ≠ ∞)).mp (hloc (F x t)) have hd : ContinuousAt (fun s => (F x s,t-s)) t := ((hc x).continuousAt).prodMk (continuousAt_const.sub continuousAt_id) have hn : {s : ℝ | ContMDiffAt (I.prod 𝓘(ℝ,ℝ)) I 1 (Function.uncurry F) (F x s,t-s)} ∈ 𝓝 t := by have hh' : ∀ᶠ q in 𝓝 (F x t,t-t), ContMDiffAt (I.prod 𝓘(ℝ,ℝ)) I 1 (Function.uncurry F) q := by simpa only [sub_self] using hh exact hd.eventually hh' obtain ⟨s,hs,hst⟩ := mem_closure_iff_nhds.mp ht _ hn exact hstep hst hs have hZo : IsOpen Z := by apply isOpen_iff_mem_nhds.mpr intro s hs have hh := (contMDiffAt_iff_contMDiffAt_nhds (by simp : (1:ℕ∞ω) ≠ ∞)).mp (hloc (F x s)) have hd : ContinuousAt (fun t : ℝ => (F x s,t-s)) s := continuousAt_const.prodMk (continuousAt_id.sub continuousAt_const) have hn : {t : ℝ | ContMDiffAt (I.prod 𝓘(ℝ,ℝ)) I 1 (Function.uncurry F) (F x s,t-s)} ∈ 𝓝 s := by have hh' : ∀ᶠ q in 𝓝 (F x s,s-s), ContMDiffAt (I.prod 𝓘(ℝ,ℝ)) I 1 (Function.uncurry F) q := by simpa only [sub_self] using hh exact hd.eventually hh' exact Filter.mem_of_superset hn (fun t ht => hstep hs ht) have h0Z : (0:ℝ) ∈ Z := contMDiffAt_id.congr_of_eventuallyEq (Filter.Eventually.of_forall h0) have he := (show IsClopen Z from ⟨hZc,hZo⟩).eq_univ ⟨0,h0Z⟩ change t ∈ Z rw [he] trivial intro p have hfirst : ContMDiffAt (I.prod 𝓘(ℝ,ℝ)) I 1 (fun q : X × ℝ => F q.1 p.2) p := (htime p.1 p.2).comp p contMDiffAt_fst have hsecond : ContMDiffAt (I.prod 𝓘(ℝ,ℝ)) 𝓘(ℝ,ℝ) 1 (fun q : X × ℝ => q.2-p.2) p := contMDiffAt_snd.sub contMDiffAt_const have hl : ContMDiffAt (I.prod 𝓘(ℝ,ℝ)) I 1 (Function.uncurry F) (F p.1 p.2,p.2-p.2) := by simpa only [sub_self] using hloc (F p.1 p.2) apply (hl.comp p (hfirst.prodMk hsecond)).congr_of_eventuallyEq exact Filter.Eventually.of_forall (fun q => by change F q.1 q.2 = F (F q.1 p.2) (q.2-p.2) rw [hadd,add_sub_cancel]) end ThreeManifold.FlowContinuation end section open Set Filter Topology Bundle Manifold IsManifold open scoped ContDiff namespace ThreeManifold.RiemannianCoordinates section variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] [CompactSpace M] [T2Space M] def globalGeodesic (g : SmoothMetric M) (p : TangentBundle I3 M) : ℝ → M := (exists_global_geodesic g p).choose lemma globalGeodesic_initial (g : SmoothMetric M) (p : TangentBundle I3 M) : phaseLift (globalGeodesic g p) 0 = p := (exists_global_geodesic g p).choose_spec.1 lemma globalGeodesic_isGeodesic (g : SmoothMetric M) (p : TangentBundle I3 M) : IsGeodesicOn g (globalGeodesic g p) univ := (exists_global_geodesic g p).choose_spec.2 def geodesicFlow (g : SmoothMetric M) (p : TangentBundle I3 M) (t : ℝ) : TangentBundle I3 M := phaseLift (globalGeodesic g p) t lemma geodesicFlow_zero (g : SmoothMetric M) (p : TangentBundle I3 M) : geodesicFlow g p 0 = p := globalGeodesic_initial g p lemma geodesicFlow_continuous (g : SmoothMetric M) (p : TangentBundle I3 M) : Continuous (geodesicFlow g p) := continuous_iff_continuousAt.mpr (fun t => phaseLift_continuousAt isOpen_univ (globalGeodesic_isGeodesic g p).1 (mem_univ t)) lemma geodesicFlow_add (g : SmoothMetric M) (p : TangentBundle I3 M) (s t : ℝ) : geodesicFlow g (geodesicFlow g p s) t = geodesicFlow g p (s+t) := by have hγ := globalGeodesic_isGeodesic g p have hη := globalGeodesic_isGeodesic g (geodesicFlow g p s) have htrans : IsGeodesicOn g (fun r => globalGeodesic g p (r+s)) univ := by simpa using geodesic_translate g isOpen_univ hγ s have h0 : phaseLift (globalGeodesic g (geodesicFlow g p s)) 0 = phaseLift (fun r => globalGeodesic g p (r+s)) 0 := by rw [globalGeodesic_initial,phaseLift_translate (hγ.1.contMDiffAt (by simp)),zero_add] rfl have he := geodesic_unique g isOpen_univ ordConnected_univ hη htrans (mem_univ 0) h0 have he' := phaseLift_eventuallyEq (t := t) (Filter.Eventually.of_forall (fun r => he (mem_univ r))) change phaseLift _ t = _ rw [he'.eq_of_nhds,phaseLift_translate (hγ.1.contMDiffAt (by simp)),add_comm] rfl lemma geodesicFlow_eq_chart (g : SmoothMetric M) (p : TangentBundle I3 M) {ε : ℝ} (hε : 0 < ε) {X : (E3 × E3) → ℝ → E3 × E3} {y : TangentBundle I3 M} (hy : y ∈ (chartAt (ModelProd E3 E3) p).source) (h0 : X (chartAt (ModelProd E3 E3) p y) 0 = chartAt (ModelProd E3 E3) p y) (hd : ∀ t ∈ Ioo (-ε) ε, HasDerivAt (X (chartAt (ModelProd E3 E3) p y)) (geodesicPhase g (chartAt E3 p.1) (X (chartAt (ModelProd E3 E3) p y) t)) t) (hXt : ∀ t ∈ Ioo (-ε) ε, (X (chartAt (ModelProd E3 E3) p y) t).1 ∈ (chartAt E3 p.1).target) {t : ℝ} (ht : t ∈ Ioo (-ε) ε) : geodesicFlow g y t = (extChartAt I3.tangent p).symm (X (chartAt (ModelProd E3 E3) p y) t) := by let c := chartAt E3 p.1 let C := chartAt (ModelProd E3 E3) p let γ : ℝ → M := fun r => c.symm (X (C y) r).1 have hγ : IsGeodesicOn g γ (Ioo (-ε) ε) := chart_geodesic g (chart_mem_atlas E3 p.1) isOpen_Ioo hXt hd have hz : (0:ℝ) ∈ Ioo (-ε) ε := ⟨by linarith,hε⟩ have hphase : ∀ r ∈ Ioo (-ε) ε, C (phaseLift γ r) = X (C y) r := by intro r hr rw [phaseLift_chart (hγ.1.contMDiffAt (isOpen_Ioo.mem_nhds hr)) (c.map_target (hXt r hr))] have he : c ∘ γ =ᶠ[𝓝 r] fun t => (X (C y) t).1 := Filter.eventually_of_mem (isOpen_Ioo.mem_nhds hr) (fun t ht => c.right_inv (hXt t ht)) apply Prod.ext · exact he.eq_of_nhds · have hh : HasDerivAt (fun t => (X (C y) t).1) (X (C y) r).2 r := (hd r hr).fst exact he.deriv_eq.trans hh.deriv have hinit : phaseLift γ 0 = y := C.injOn ((TangentBundle.mem_chart_source_iff _ _).mpr (c.map_target (hXt 0 hz))) hy ((hphase 0 hz).trans h0) have he := geodesic_unique g isOpen_Ioo ordConnected_Ioo hγ (show IsGeodesicOn g (globalGeodesic g y) (Ioo (-ε) ε) from ⟨(globalGeodesic_isGeodesic g y).1.mono (subset_univ _), fun c hc t ht hct => (globalGeodesic_isGeodesic g y).2 c hc t (mem_univ t) hct⟩) hz (hinit.trans (globalGeodesic_initial g y).symm) have het : phaseLift γ t = geodesicFlow g y t := (phaseLift_eventuallyEq (Filter.eventually_of_mem (isOpen_Ioo.mem_nhds ht) (fun r hr => he hr))).eq_of_nhds rw [← het,← hphase t ht] exact (C.left_inv ((TangentBundle.mem_chart_source_iff _ _).mpr (c.map_target (hXt t ht)))).symm lemma geodesicFlow_contMDiffAt_zero (g : SmoothMetric M) (p : TangentBundle I3 M) : ContMDiffAt (I3.tangent.prod 𝓘(ℝ,ℝ)) I3.tangent 1 (Function.uncurry (geodesicFlow g)) (p,0) := by let c := chartAt E3 p.1 let C := chartAt (ModelProd E3 E3) p let e := extChartAt I3.tangent p have hp : p ∈ C.source := mem_chart_source _ p obtain ⟨r,hr,ε,hε,X,h0,hγ,hd,hXt,hXs⟩ := exists_geodesic_local_flow g (z := e p) (chart_mem_atlas E3 p.1) (c.map_source (mem_chart_source E3 p.1)) have hz : (0:ℝ) ∈ Ioo (-ε) ε := ⟨by linarith,hε⟩ have hXp : ContDiffAt ℝ 1 (Function.uncurry X) (e p,0) := hXs.contDiffAt ((Metric.isOpen_ball.prod isOpen_Ioo).mem_nhds ⟨Metric.mem_ball_self hr,hz⟩) have he : ContMDiffAt I3.tangent 𝓘(ℝ,E3 × E3) 1 e p := contMDiffAt_extChartAt have harg : ContMDiffAt (I3.tangent.prod 𝓘(ℝ,ℝ)) 𝓘(ℝ,(E3 × E3) × ℝ) 1 (fun q : TangentBundle I3 M × ℝ => (e q.1,q.2)) (p,0) := (he.comp (p,0) contMDiffAt_fst).prodMk_space contMDiffAt_snd have hOut : ContMDiffAt 𝓘(ℝ,E3 × E3) I3.tangent 1 e.symm (X (e p) 0) := by have hh := contMDiffWithinAt_extChartAt_symm_target (I := I3.tangent) (n := 1) p (show X (e p) 0 ∈ e.target from by rw [h0 _ (Metric.mem_ball_self hr)]; exact e.map_source (mem_extChartAt_source p)) apply hh.contMDiffAt rw [h0 _ (Metric.mem_ball_self hr)] exact extChartAt_target_mem_nhds p apply (hOut.comp (p,0) (hXp.contMDiffAt.comp (p,0) harg)).congr_of_eventuallyEq have hn : {y : TangentBundle I3 M | y ∈ C.source ∧ e y ∈ Metric.ball (e p) r} ∈ 𝓝 p := inter_mem (C.open_source.mem_nhds hp) (he.continuousAt.preimage_mem_nhds (Metric.isOpen_ball.mem_nhds (Metric.mem_ball_self hr))) have ht : Ioo (-ε) ε ∈ 𝓝 (0:ℝ) := isOpen_Ioo.mem_nhds hz have hnh := prod_mem_nhds hn ht filter_upwards [hnh] with q hq exact geodesicFlow_eq_chart g p hε hq.1.1 (h0 _ hq.1.2) (hd _ hq.1.2) (hXt _ hq.1.2) hq.2 theorem geodesicFlow_contMDiff (g : SmoothMetric M) : ContMDiff (I3.tangent.prod 𝓘(ℝ,ℝ)) I3.tangent 1 (Function.uncurry (geodesicFlow g)) := FlowContinuation.flow_contMDiff (geodesicFlow g) (geodesicFlow_zero g) (geodesicFlow_add g) (geodesicFlow_continuous g) (geodesicFlow_contMDiffAt_zero g) end section variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] theorem exists_geodesic_local_variation (g : SmoothMetric M) {c : OpenPartialHomeomorph M E3} (hc : c ∈ atlas E3 M) {z : E3 × E3} (hz : z.1 ∈ c.target) : ∃ r > (0 : ℝ), ∃ ε > (0 : ℝ), ∃ X : (E3 × E3) → ℝ → E3 × E3, (∀ y ∈ Metric.ball z r, X y 0 = y) ∧ (∀ y ∈ Metric.ball z r, IsGeodesicOn g (fun t => c.symm (X y t).1) (Ioo (-ε) ε)) ∧ (∀ y ∈ Metric.ball z r, ∀ t ∈ Ioo (-ε) ε, HasDerivAt (X y) (geodesicPhase g c (X y t)) t) ∧ (∀ y ∈ Metric.ball z r, ∀ t ∈ Ioo (-ε) ε, (X y t).1 ∈ c.target) ∧ ContDiffOn ℝ 1 (Function.uncurry X) (Metric.ball z r ×ˢ Ioo (-ε) ε) ∧ ∃ A : (E3 × E3) → ℝ → (E3 × E3) →L[ℝ] E3 × E3, (∀ y ∈ Metric.ball z r, A y 0 = ContinuousLinearMap.id ℝ (E3 × E3)) ∧ (∀ y ∈ Metric.ball z r, ∀ t ∈ Ioo (-ε) ε, HasFDerivAt (fun w => X w t) (A y t) y) ∧ (∀ y ∈ Metric.ball z r, ∀ t ∈ Ioo (-ε) ε, HasDerivAt (A y) ((fderiv ℝ (geodesicPhase g c) (X y t)).comp (A y t)) t) := by obtain ⟨r,hr,ε,hε,X,h0,hd,hs,A,hA0,hdf,hAd⟩ := ODEVariation.exists_contDiff_variation_flow (contDiffAt_infty.mp (geodesicPhase_smoothAt g hc hz) 2) have hO : IsOpen (Metric.ball z r ×ˢ Ioo (-ε) ε) := Metric.isOpen_ball.prod isOpen_Ioo have hp : (z,(0:ℝ)) ∈ Metric.ball z r ×ˢ Ioo (-ε) ε := ⟨Metric.mem_ball_self hr,by constructor <;> linarith⟩ have hx : ContinuousAt (Function.uncurry X) (z,0) := hs.continuousOn.continuousAt (hO.mem_nhds hp) have hmem : {p : (E3 × E3) × ℝ | (X p.1 p.2).1 ∈ c.target} ∈ 𝓝 (z,0) := by apply hx.fst.preimage_mem_nhds simpa only [Function.uncurry_apply_pair,h0 z (Metric.mem_ball_self hr)] using c.open_target.mem_nhds hz obtain ⟨η,hη,hsmall⟩ := Metric.mem_nhds_iff.mp (inter_mem hmem (hO.mem_nhds hp)) have hsub : ∀ y ∈ Metric.ball z (η/2), ∀ t ∈ Ioo (-(η/2)) (η/2), (X y t).1 ∈ c.target ∧ (y,t) ∈ Metric.ball z r ×ˢ Ioo (-ε) ε := by intro y hy t ht apply hsmall (a := (y,t)) rw [Metric.mem_ball,Prod.dist_eq,max_lt_iff] exact ⟨lt_trans hy (half_lt_self hη),by rw [Real.dist_eq,sub_zero,abs_lt]; constructor <;> linarith [ht.1,ht.2]⟩ refine ⟨η/2,half_pos hη,η/2,half_pos hη,X,?_,?_,?_,?_,?_⟩ · intro y hy exact h0 y (hsub y hy 0 ⟨by linarith,by linarith⟩).2.1 · intro y hy exact chart_geodesic g hc isOpen_Ioo (fun t ht => (hsub y hy t ht).1) (fun t ht => hd y (hsub y hy t ht).2.1 t (hsub y hy t ht).2.2) · intro y hy t ht exact hd y (hsub y hy t ht).2.1 t (hsub y hy t ht).2.2 · exact fun y hy t ht => (hsub y hy t ht).1 · refine ⟨hs.mono (fun p hp => (hsub p.1 hp.1 p.2 hp.2).2),A,?_,?_,?_⟩ · intro y hy exact hA0 y (hsub y hy 0 ⟨by linarith,by linarith⟩).2.1 · intro y hy t ht exact hdf y (hsub y hy t ht).2.1 t (hsub y hy t ht).2.2 · intro y hy t ht exact hAd y (hsub y hy t ht).2.1 t (hsub y hy t ht).2.2 lemma geodesicPhase_fderiv_apply (g : SmoothMetric M) {c : OpenPartialHomeomorph M E3} (hc : c ∈ atlas E3 M) {z : E3 × E3} (hz : z.1 ∈ c.target) (w : E3 × E3) : fderiv ℝ (geodesicPhase g c) z w = (w.2,-(fderiv ℝ (connectionBilin g c) z.1 w.1 z.2 z.2 + connectionBilin g c z.1 w.2 z.2 + connectionBilin g c z.1 z.2 w.2)) := by let : NormedAddCommGroup (E3 →L[ℝ] E3) := ContinuousLinearMap.toNormedAddCommGroup let : NormedSpace ℝ (E3 →L[ℝ] E3) := ContinuousLinearMap.toNormedSpace let : NormedAddCommGroup (E3 →L[ℝ] E3 →L[ℝ] E3) := ContinuousLinearMap.toNormedAddCommGroup let : NormedSpace ℝ (E3 →L[ℝ] E3 →L[ℝ] E3) := ContinuousLinearMap.toNormedSpace have hC := ((connectionBilin_smoothAt g hc hz).differentiableAt (by simp)).hasFDerivAt.comp z (hasFDerivAt_fst (p := z)) have hh := (hasFDerivAt_snd (p := z)).prodMk (((hC.clm_apply (hasFDerivAt_snd (p := z))).clm_apply (hasFDerivAt_snd (p := z))).neg) have he : geodesicPhase g c = (fun q : E3 × E3 => (q.2,-connectionBilin g c q.1 q.2 q.2)) := by funext q simp only [geodesicPhase,connectionBilin_apply] change HasFDerivAt (fun q : E3 × E3 => (q.2,-connectionBilin g c q.1 q.2 q.2)) _ z at hh rw [he,hh.fderiv] simp only [ContinuousLinearMap.prod_apply, _root_.neg_apply,_root_.add_apply,ContinuousLinearMap.comp_apply, ContinuousLinearMap.flip_apply,Function.comp_apply] congr 1 change -(connectionBilin g c z.1 z.2 w.2 + (connectionBilin g c z.1 w.2 z.2 + fderiv ℝ (connectionBilin g c) z.1 w.1 z.2 z.2)) = _ abel lemma geodesicPhase_variation_smooth (g : SmoothMetric M) {c : OpenPartialHomeomorph M E3} (hc : c ∈ atlas E3 M) {U : Set ℝ} (hU : IsOpen U) {z w : ℝ → E3 × E3} (hz : ∀ t ∈ U, (z t).1 ∈ c.target) (hdz : ∀ t ∈ U, HasDerivAt z (geodesicPhase g c (z t)) t) (hdw : ∀ t ∈ U, HasDerivAt w (fderiv ℝ (geodesicPhase g c) (z t) (w t)) t) : ContDiffOn ℝ ∞ (fun t => (z t,w t)) U := by apply ODEVariation.solution_smooth (f := fun q : (E3 × E3) × (E3 × E3) => (geodesicPhase g c q.1,fderiv ℝ (geodesicPhase g c) q.1 q.2)) (S := {q | q.1.1 ∈ c.target}) _ hU (fun t ht => hz t ht) (fun t ht => (hdz t ht).prodMk (hdw t ht)) intro q hq have hf := geodesicPhase_smoothAt g hc hq exact ((hf.comp q contDiffAt_fst).prodMk ((hf.fderiv_right (m := ∞) (by simp)).comp q contDiffAt_fst |>.clm_apply contDiffAt_snd)).contDiffWithinAt lemma linearized_geodesic_jacobi (g : SmoothMetric M) {c : OpenPartialHomeomorph M E3} (hc : c ∈ atlas E3 M) {U : Set ℝ} (hU : IsOpen U) {z w : ℝ → E3 × E3} (hz : ∀ t ∈ U, (z t).1 ∈ c.target) (hdz : ∀ t ∈ U, HasDerivAt z (geodesicPhase g c (z t)) t) (hdw : ∀ t ∈ U, HasDerivAt w (fderiv ℝ (geodesicPhase g c) (z t) (w t)) t) {s : ℝ} (hs : s ∈ U) : alongDerivative g c (fun t => (z t).1) (alongDerivative g c (fun t => (z t).1) (fun t => (w t).1)) s + curvatureTerm g c (z s).1 (w s).1 (z s).2 (z s).2 = 0 := by let : NormedAddCommGroup (E3 →L[ℝ] E3) := ContinuousLinearMap.toNormedAddCommGroup let : NormedSpace ℝ (E3 →L[ℝ] E3) := ContinuousLinearMap.toNormedSpace let : NormedAddCommGroup (E3 →L[ℝ] E3 →L[ℝ] E3) := ContinuousLinearMap.toNormedAddCommGroup let : NormedSpace ℝ (E3 →L[ℝ] E3 →L[ℝ] E3) := ContinuousLinearMap.toNormedSpace let C := connectionBilin g c have hdw' (t : ℝ) (ht : t ∈ U) := (hdw t ht).congr_deriv (geodesicPhase_fderiv_apply g hc (hz t ht) (w t)) have hx (t : ℝ) (ht : t ∈ U) : HasDerivAt (fun r => (z r).1) (z t).2 t := (hdz t ht).fst have hv (t : ℝ) (ht : t ∈ U) : HasDerivAt (fun r => (z r).2) (-C (z t).1 (z t).2 (z t).2) t := by have hh : HasDerivAt (fun r => (z r).2) (geodesicPhase g c (z t)).2 t := (hdz t ht).snd simpa only [C,connectionBilin_apply,geodesicPhase] using hh have hj (t : ℝ) (ht : t ∈ U) : HasDerivAt (fun r => (w r).1) (w t).2 t := (hdw' t ht).fst have hk (t : ℝ) (ht : t ∈ U) : HasDerivAt (fun r => (w r).2) (-(fderiv ℝ C (z t).1 (w t).1 (z t).2 (z t).2 + C (z t).1 (w t).2 (z t).2 + C (z t).1 (z t).2 (w t).2)) t := (hdw' t ht).snd let F : ℝ → E3 := fun t => (w t).2 + C (z t).1 (z t).2 (w t).1 have he : alongDerivative g c (fun t => (z t).1) (fun t => (w t).1) =ᶠ[𝓝 s] F := by filter_upwards [hU.mem_nhds hs] with t ht rw [alongDerivative,(hx t ht).deriv,(hj t ht).deriv] exact (connectionBilin_apply g c (z t).1 (z t).2 (w t).1).symm ▸ rfl have hC := ((connectionBilin_smoothAt g hc (hz s hs)).differentiableAt (by simp)).hasFDerivAt.comp_hasDerivAt s (hx s hs) have hF := (hk s hs).add ((hC.clm_apply (hv s hs)).clm_apply (hj s hs)) change HasDerivAt F _ s at hF rw [along_congr g c Filter.EventuallyEq.rfl he,alongDerivative,(hx s hs).deriv,hF.deriv, curvature_connection g hc (hz s hs)] rw [← connectionBilin_apply] change -(fderiv ℝ C (z s).1 (w s).1 (z s).2 (z s).2 + C (z s).1 (w s).2 (z s).2 + C (z s).1 (z s).2 (w s).2) + ((fderiv ℝ C (z s).1 (z s).2 (z s).2 + C (z s).1 (-C (z s).1 (z s).2 (z s).2)) (w s).1 + C (z s).1 (z s).2 (w s).2) + C (z s).1 (z s).2 ((w s).2 + C (z s).1 (z s).2 (w s).1) + (fderiv ℝ C (z s).1 (w s).1 (z s).2 (z s).2 - fderiv ℝ C (z s).1 (z s).2 (w s).1 (z s).2 + C (z s).1 (w s).1 (C (z s).1 (z s).2 (z s).2) - C (z s).1 (z s).2 (C (z s).1 (w s).1 (z s).2)) = 0 simp only [_root_.add_apply,map_neg,_root_.neg_apply,map_add] rw [connectionBilin_deriv_symm g hc (hz s hs) (z s).2 (z s).2 (w s).1, connectionBilin_symm g c (z s).1 (w s).2 (z s).2, connectionBilin_symm g c (z s).1 (C (z s).1 (z s).2 (z s).2) (w s).1, connectionBilin_symm g c (z s).1 (w s).1 (z s).2] abel end section variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] variable [CompactSpace M] [T2Space M] variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] lemma geodesic_family_contMDiffAt (g : SmoothMetric M) {f : E → TangentBundle I3 M} {a : E} (hf : ContMDiffAt 𝓘(ℝ,E) I3.tangent 1 f a) (s : ℝ) : ContMDiffAt 𝓘(ℝ,E) I3.tangent 1 (fun y => geodesicFlow g (f y) s) a := by exact (geodesicFlow_contMDiff g).contMDiffAt.comp a (hf.prodMk contMDiffAt_const) theorem geodesic_variation_representation (g : SmoothMetric M) {f : E → TangentBundle I3 M} {a : E} (hf : ContMDiffAt 𝓘(ℝ,E) I3.tangent 1 f a) (s : ℝ) : let p := geodesicFlow g (f a) s let e := extChartAt I3.tangent p let Z := fun y => e (geodesicFlow g (f y) s) ∃ ε > (0 : ℝ), ∃ X : (E3 × E3) → ℝ → E3 × E3, ∃ A : (E3 × E3) → ℝ → (E3 × E3) →L[ℝ] E3 × E3, (∀ t ∈ Ioo (-ε) ε, e (geodesicFlow g (f a) (s+t)) = X (Z a) t) ∧ (∀ t ∈ Ioo (-ε) ε, HasDerivAt (X (Z a)) (geodesicPhase g (chartAt E3 p.1) (X (Z a) t)) t) ∧ (∀ t ∈ Ioo (-ε) ε, (X (Z a) t).1 ∈ (chartAt E3 p.1).target) ∧ (∀ t ∈ Ioo (-ε) ε, HasFDerivAt (fun y => e (geodesicFlow g (f y) (s+t))) ((A (Z a) t).comp (fderiv ℝ Z a)) a) ∧ (∀ t ∈ Ioo (-ε) ε, HasDerivAt (A (Z a)) ((fderiv ℝ (geodesicPhase g (chartAt E3 p.1)) (X (Z a) t)).comp (A (Z a) t)) t) := by intro p e Z let c := chartAt E3 p.1 let C := chartAt (ModelProd E3 E3) p have hp : p ∈ C.source := mem_chart_source _ p have hz : (e p).1 ∈ c.target := c.map_source (mem_chart_source E3 p.1) obtain ⟨r,hr,ε,hε,X,h0,hγ,hd,hXt,hXs,A,hA0,hdf,hAd⟩ := exists_geodesic_local_variation g (chart_mem_atlas E3 p.1) hz have ha : Z a ∈ Metric.ball (e p) r := Metric.mem_ball_self hr have hFa := geodesic_family_contMDiffAt g hf s have he : ContMDiffAt I3.tangent 𝓘(ℝ,E3 × E3) 1 e p := contMDiffAt_extChartAt have hZ : ContDiffAt ℝ 1 Z a := contMDiffAt_iff_contDiffAt.mp (he.comp a hFa) have hZn : {y : E | geodesicFlow g (f y) s ∈ C.source ∧ Z y ∈ Metric.ball (e p) r} ∈ 𝓝 a := inter_mem (hFa.continuousAt.preimage_mem_nhds (C.open_source.mem_nhds hp)) (hZ.continuousAt.preimage_mem_nhds (Metric.isOpen_ball.mem_nhds ha)) have hrep (t : ℝ) (ht : t ∈ Ioo (-ε) ε) : (fun y => e (geodesicFlow g (f y) (s+t))) =ᶠ[𝓝 a] fun y => X (Z y) t := by filter_upwards [hZn] with y hy rw [← geodesicFlow_add g (f y) s t,geodesicFlow_eq_chart g p hε hy.1 (h0 _ hy.2) (hd _ hy.2) (hXt _ hy.2) ht] apply e.right_inv change X (Z y) t ∈ (extChartAt I3.tangent p).target rw [extChartAt_target] exact ⟨(TangentBundle.mem_chart_target_iff _ _).mpr (hXt _ hy.2 t ht),by simp⟩ refine ⟨ε,hε,X,A,fun t ht => (hrep t ht).eq_of_nhds,hd _ ha,hXt _ ha,?_,hAd _ ha⟩ intro t ht exact ((hdf _ ha t ht).comp a (hZ.differentiableAt (by simp)).hasFDerivAt).congr_of_eventuallyEq (hrep t ht) theorem geodesic_family_linearization (g : SmoothMetric M) {f : E → TangentBundle I3 M} {a : E} (hf : ContMDiffAt 𝓘(ℝ,E) I3.tangent 1 f a) (s : ℝ) (b : E) : let p := geodesicFlow g (f a) s let e := extChartAt I3.tangent p let z := fun t => e (geodesicFlow g (f a) (s+t)) let w := fun t => fderiv ℝ (fun y => e (geodesicFlow g (f y) (s+t))) a b ∃ ε > (0 : ℝ), (∀ t ∈ Ioo (-ε) ε, (z t).1 ∈ (chartAt E3 p.1).target) ∧ (∀ t ∈ Ioo (-ε) ε, HasDerivAt z (geodesicPhase g (chartAt E3 p.1) (z t)) t) ∧ (∀ t ∈ Ioo (-ε) ε, HasDerivAt w (fderiv ℝ (geodesicPhase g (chartAt E3 p.1)) (z t) (w t)) t) := by intro p e z w obtain ⟨ε,hε,X,A,hX,hd,hXt,hdf,hAd⟩ := geodesic_variation_representation g hf s let Z := fun y => e (geodesicFlow g (f y) s) have hew : ∀ t ∈ Ioo (-ε) ε, w t = A (Z a) t (fderiv ℝ Z a b) := by intro t ht exact congrArg (fun L => L b) (hdf t ht).fderiv have hen (t : ℝ) (ht : t ∈ Ioo (-ε) ε) : z =ᶠ[𝓝 t] X (Z a) := Filter.eventually_of_mem (isOpen_Ioo.mem_nhds ht) (fun r hr => hX r hr) have hwen (t : ℝ) (ht : t ∈ Ioo (-ε) ε) : w =ᶠ[𝓝 t] fun r => A (Z a) r (fderiv ℝ Z a b) := Filter.eventually_of_mem (isOpen_Ioo.mem_nhds ht) (fun r hr => hew r hr) refine ⟨ε,hε,?_,?_,?_⟩ · intro t ht rw [show z t = X (Z a) t from hX t ht] exact hXt t ht · intro t ht rw [(hen t ht).eq_of_nhds] exact (hd t ht).congr_of_eventuallyEq (hen t ht) · intro t ht rw [(hen t ht).eq_of_nhds,(hwen t ht).eq_of_nhds] have hh := ((hAd t ht).clm_apply (hasDerivAt_const t (fderiv ℝ Z a b))).congr_of_eventuallyEq (hwen t ht) simpa only [map_zero,add_zero,ContinuousLinearMap.comp_apply] using hh lemma geodesic_family_linearization_at (g : SmoothMetric M) {f : E → TangentBundle I3 M} {a : E} (hf : ContMDiffAt 𝓘(ℝ,E) I3.tangent 1 f a) (s : ℝ) (b : E) : let p := geodesicFlow g (f a) s let e := extChartAt I3.tangent p let z := fun t => e (geodesicFlow g (f a) t) let w := fun t => fderiv ℝ (fun y => e (geodesicFlow g (f y) t)) a b ∃ U : Set ℝ, IsOpen U ∧ s ∈ U ∧ (∀ t ∈ U, (z t).1 ∈ (chartAt E3 p.1).target) ∧ (∀ t ∈ U, HasDerivAt z (geodesicPhase g (chartAt E3 p.1) (z t)) t) ∧ (∀ t ∈ U, HasDerivAt w (fderiv ℝ (geodesicPhase g (chartAt E3 p.1)) (z t) (w t)) t) := by intro p e z w obtain ⟨ε,hε,hz,hdz,hdw⟩ := geodesic_family_linearization g hf s b have hh (t : ℝ) (ht : t ∈ Ioo (s-ε) (s+ε)) : t-s ∈ Ioo (-ε) ε := by constructor <;> linarith [ht.1,ht.2] refine ⟨Ioo (s-ε) (s+ε),isOpen_Ioo,by constructor <;> linarith,?_,?_,?_⟩ · intro t ht simpa only [add_sub_cancel] using hz (t-s) (hh t ht) · intro t ht have hd := (hdz (t-s) (hh t ht)).scomp t ((hasDerivAt_id t).sub_const s) simpa only [Function.comp_def,add_sub_cancel,mul_one,one_smul,id_eq,z,w,p,e] using hd · intro t ht have hd := (hdw (t-s) (hh t ht)).scomp t ((hasDerivAt_id t).sub_const s) simpa only [Function.comp_def,add_sub_cancel,mul_one,one_smul,id_eq,z,w,p,e] using hd end section variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] lemma smoothField_of_local_coordinates {γ : ℝ → M} {J : ∀ s, TangentSpace I3 (γ s)} {U : Set ℝ} (hU : IsOpen U) (hγ : ContMDiffOn 𝓘(ℝ,ℝ) I3 ∞ γ U) (h : ∀ s ∈ U, ∃ c : OpenPartialHomeomorph M E3, c ∈ atlas E3 M ∧ γ s ∈ c.source ∧ ContDiffAt ℝ ∞ (fieldCoordinates γ J c) s) : SmoothFieldOn γ J U := by intro d hd s hs obtain ⟨c,hc,hsc,hF⟩ := h s hs.1 have hγs := hγ.contMDiffAt (hU.mem_nhds hs.1) have fsm := transition_smoothAt hc hd (c.map_source hsc) (show c.symm (c (γ s)) ∈ d.source by rw [c.left_inv hsc]; exact hs.2) have hx := curve_coordinates_smoothAt hU hγ hc hs.1 hsc have he : fieldCoordinates γ J d =ᶠ[𝓝 s] fun r => fderiv ℝ (d ∘ c.symm) (c (γ r)) (fieldCoordinates γ J c r) := by filter_upwards [hγs.continuousAt.preimage_mem_nhds ((c.open_source.inter d.open_source).mem_nhds ⟨hsc,hs.2⟩)] with r hr exact (field_coordinates_change hc hd hr.1 hr.2).symm exact (((fsm.fderiv_right (m := ∞) (by simp)).comp s hx).clm_apply hF).congr_of_eventuallyEq he |>.contDiffWithinAt lemma jacobi_of_local_coordinates (g : SmoothMetric M) {γ : ℝ → M} {J : ∀ s, TangentSpace I3 (γ s)} {U : Set ℝ} (hU : IsOpen U) (hγ : ContMDiffOn 𝓘(ℝ,ℝ) I3 ∞ γ U) (h : ∀ s ∈ U, ∃ c : OpenPartialHomeomorph M E3, c ∈ atlas E3 M ∧ γ s ∈ c.source ∧ ContDiffAt ℝ ∞ (fieldCoordinates γ J c) s ∧ alongDerivative g c (c ∘ γ) (alongDerivative g c (c ∘ γ) (fieldCoordinates γ J c)) s + curvatureTerm g c (c (γ s)) (fieldCoordinates γ J c s) (deriv (c ∘ γ) s) (deriv (c ∘ γ) s) = 0) : IsJacobiFieldOn g γ J U := by have hJ : SmoothFieldOn γ J U := smoothField_of_local_coordinates hU hγ (fun s hs => by obtain ⟨c,hc,hsc,hcs,_⟩ := h s hs; exact ⟨c,hc,hsc,hcs⟩) have hi (s : ℝ) (hs : s ∈ U) : covariantDerivative g γ (covariantDerivative g γ J) s + curvatureAlong g γ J s = 0 := by obtain ⟨c,hc,hsc,_,he⟩ := h s hs apply (mdifferentiable_of_mem_atlas (I := I3) hc).mfderiv_injective hsc rw [map_add,map_zero] change fieldCoordinates γ (covariantDerivative g γ (covariantDerivative g γ J)) c s + fieldCoordinates γ (curvatureAlong g γ J) c s = 0 rw [covariantDerivative_iterate_coordinates g hU hγ hJ hs hc hsc, curvatureAlong_coordinates g J (hγ.contMDiffAt (hU.mem_nhds hs)) hc hsc] exact he intro c hc refine ⟨hJ c hc,?_⟩ intro s hs hsc rw [← covariantDerivative_iterate_coordinates g hU hγ hJ hs hc hsc, ← curvatureAlong_coordinates g J (hγ.contMDiffAt (hU.mem_nhds hs)) hc hsc] change mfderiv I3 I3 c (γ s) (covariantDerivative g γ (covariantDerivative g γ J) s) + mfderiv I3 I3 c (γ s) (curvatureAlong g γ J s) = 0 rw [← map_add,hi s hs,map_zero] end variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] variable [CompactSpace M] [T2Space M] variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] def geodesicFamily (g : SmoothMetric M) (f : E → TangentBundle I3 M) (a : E) (t : ℝ) : M := (geodesicFlow g (f a) t).proj def geodesicVariation (g : SmoothMetric M) (f : E → TangentBundle I3 M) (a b : E) (t : ℝ) : TangentSpace I3 (geodesicFamily g f a t) := mfderiv 𝓘(ℝ,E) I3 (fun y => geodesicFamily g f y t) a b lemma geodesicFamily_isGeodesic (g : SmoothMetric M) (f : E → TangentBundle I3 M) (a : E) : IsGeodesicOn g (geodesicFamily g f a) univ := by let _ : NormedAddCommGroup E := inferInstance let _ : NormedSpace ℝ E := inferInstance exact globalGeodesic_isGeodesic g (f a) lemma geodesicFamily_parameter_smooth (g : SmoothMetric M) {f : E → TangentBundle I3 M} {a : E} (hf : ContMDiffAt 𝓘(ℝ,E) I3.tangent 1 f a) (t : ℝ) : ContMDiffAt 𝓘(ℝ,E) I3 1 (fun y => geodesicFamily g f y t) a := (contMDiff_proj (IB := I3) (F := E3) (E := TangentSpace I3)).contMDiffAt.comp a (geodesic_family_contMDiffAt g hf t) lemma geodesicVariation_coordinates (g : SmoothMetric M) {f : E → TangentBundle I3 M} {a : E} (hf : ContMDiffAt 𝓘(ℝ,E) I3.tangent 1 f a) (b : E) (t : ℝ) {p : TangentBundle I3 M} (hp : geodesicFamily g f a t ∈ (chartAt E3 p.1).source) : fieldCoordinates (geodesicFamily g f a) (geodesicVariation g f a b) (chartAt E3 p.1) t = (fderiv ℝ (fun y => extChartAt I3.tangent p (geodesicFlow g (f y) t)) a b).1 := by let c := chartAt E3 p.1 let e := extChartAt I3.tangent p have ha := geodesic_family_contMDiffAt g hf t have hb := geodesicFamily_parameter_smooth g hf t have he : ContMDiffAt I3.tangent 𝓘(ℝ,E3 × E3) 1 e (geodesicFlow g (f a) t) := contMDiffAt_extChartAt' ((TangentBundle.mem_chart_source_iff _ _).mpr hp) have hs : ContDiffAt ℝ 1 (fun y => e (geodesicFlow g (f y) t)) a := contMDiffAt_iff_contDiffAt.mp (he.comp a ha) have hd : HasFDerivAt (fun y => (e (geodesicFlow g (f y) t)).1) ((ContinuousLinearMap.fst ℝ E3 E3).comp (fderiv ℝ (fun y => e (geodesicFlow g (f y) t)) a)) a := (hs.differentiableAt (by simp)).hasFDerivAt.fst have hm := mfderiv_comp a ((mdifferentiable_of_mem_atlas (I := I3) (chart_mem_atlas E3 p.1)).mdifferentiableAt hp) (hb.mdifferentiableAt (by simp)) have hv := congrArg (fun L => L b) hm rw [mfderiv_eq_fderiv] at hv change fderiv ℝ (fun y => c (geodesicFamily g f y t)) a b = fieldCoordinates (geodesicFamily g f a) (geodesicVariation g f a b) c t at hv rw [← hv] change fderiv ℝ (fun y => (e (geodesicFlow g (f y) t)).1) a b = _ rw [hd.fderiv] rfl theorem geodesicVariation_isJacobi (g : SmoothMetric M) {f : E → TangentBundle I3 M} {a : E} (hf : ContMDiffAt 𝓘(ℝ,E) I3.tangent 1 f a) (b : E) : IsJacobiFieldOn g (geodesicFamily g f a) (geodesicVariation g f a b) univ := by let γ := geodesicFamily g f a let J := geodesicVariation g f a b have hγ := geodesicFamily_isGeodesic g f a apply jacobi_of_local_coordinates g isOpen_univ hγ.1 intro s _ let p := geodesicFlow g (f a) s let c := chartAt E3 p.1 let e := extChartAt I3.tangent p let z := fun t => e (geodesicFlow g (f a) t) let w := fun t => fderiv ℝ (fun y => e (geodesicFlow g (f y) t)) a b have hc : c ∈ atlas E3 M := chart_mem_atlas E3 p.1 have hsc : γ s ∈ c.source := mem_chart_source E3 p.1 obtain ⟨U,hU,hs,hz,hdz,hdw⟩ := geodesic_family_linearization_at g hf s b have hsmooth : ContDiffOn ℝ ∞ (fun t => (z t,w t)) U := geodesicPhase_variation_smooth g hc hU hz hdz hdw have he : fieldCoordinates γ J c =ᶠ[𝓝 s] fun t => (w t).1 := by filter_upwards [(hγ.1.contMDiffAt (by simp)).continuousAt.preimage_mem_nhds (c.open_source.mem_nhds hsc)] with t ht exact geodesicVariation_coordinates g hf b t ht have hw : ContDiffAt ℝ ∞ (fun t => (w t).1) s := (hsmooth.contDiffAt (hU.mem_nhds hs)).snd.fst refine ⟨c,hc,hsc,hw.congr_of_eventuallyEq he,?_⟩ have hde : alongDerivative g c (c ∘ γ) (fieldCoordinates γ J c) =ᶠ[𝓝 s] alongDerivative g c (fun t => (z t).1) (fun t => (w t).1) := by filter_upwards [he.eventuallyEq_nhds] with t ht exact along_congr g c Filter.EventuallyEq.rfl ht rw [along_congr g c Filter.EventuallyEq.rfl hde,he.eq_of_nhds] have hx : deriv (c ∘ γ) s = (z s).2 := by exact (show HasDerivAt (fun t => (z t).1) (z s).2 s from (hdz s hs).fst).deriv rw [hx] exact linearized_geodesic_jacobi g hc hU hz hdz hdw hs end ThreeManifold.RiemannianCoordinates end section open Set Filter Topology Bundle Manifold IsManifold Matrix MeasureTheory open scoped ContDiff Matrix.Norms.Elementwise namespace ThreeManifold.RiemannianCoordinates variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] lemma curvatureAlong_pairing_nonpos (g : SmoothMetric M) {γ : ℝ → M} (J : ∀ s, TangentSpace I3 (γ s)) {s : ℝ} (hγ : ContMDiffAt 𝓘(ℝ,ℝ) I3 ∞ γ s) (hn : NonpositiveAt g (γ s)) : g.inner (γ s) (curvatureAlong g γ J s) (J s) ≤ 0 := by have hc := chart_mem_atlas E3 (γ s) have hsc := mem_chart_source E3 (γ s) rw [← metric_field_coordinates g (curvatureAlong g γ J) J hc hsc, curvatureAlong_coordinates g J hγ hc hsc] exact hn _ hc hsc _ _ lemma npc_noConjugatePoints (g : SmoothMetric M) (hn : NonpositiveSectionalCurvature g) : NoConjugatePoints g := by apply noConjugatePoints_of_intrinsic_index g intro U hU hUc γ hγ J hJ hJc hJU apply integral_nonneg intro s by_cases hs : s ∈ U · exact indexPairDensity_nonneg_of_nonpositiveAt g J (hγ.1.contMDiffAt (hU.mem_nhds hs)) (nonpositiveAt_of_nonpositive g hn _) · have hz := notMem_subset hJU hs have hz' : indexPairDensity g γ J J s = 0 := Function.notMem_support.mp (notMem_subset (indexPairDensity_support g γ J J) hz) exact hz'.ge def jacobiNormSq (g : SmoothMetric M) (γ : ℝ → M) (J : ∀ s, TangentSpace I3 (γ s)) (s : ℝ) : ℝ := g.inner (γ s) (J s) (J s) def jacobiNormSqSlope (g : SmoothMetric M) (γ : ℝ → M) (J : ∀ s, TangentSpace I3 (γ s)) (s : ℝ) : ℝ := 2 * g.inner (γ s) (covariantDerivative g γ J s) (J s) lemma jacobiNormSq_hasDerivAt (g : SmoothMetric M) {γ : ℝ → M} {J : ∀ s, TangentSpace I3 (γ s)} {U : Set ℝ} (hU : IsOpen U) (hγ : ContMDiffOn 𝓘(ℝ,ℝ) I3 ∞ γ U) (hJ : SmoothFieldOn γ J U) {s : ℝ} (hs : s ∈ U) : HasDerivAt (jacobiNormSq g γ J) (jacobiNormSqSlope g γ J s) s := by change HasDerivAt (fun r => g.inner (γ r) (J r) (J r)) (2 * g.inner (γ s) (covariantDerivative g γ J s) (J s)) s have hd := intrinsic_pairing_hasDerivAt g hU hγ hJ hJ hs rw [g.symm (γ s) (J s) (covariantDerivative g γ J s)] at hd rw [two_mul] exact hd lemma jacobiNormSqSlope_hasDerivAt (g : SmoothMetric M) {γ : ℝ → M} {J : ∀ s, TangentSpace I3 (γ s)} {U : Set ℝ} (hU : IsOpen U) (hγ : ContMDiffOn 𝓘(ℝ,ℝ) I3 ∞ γ U) (hJ : IsJacobiFieldOn g γ J U) {s : ℝ} (hs : s ∈ U) : HasDerivAt (jacobiNormSqSlope g γ J) (2 * indexPairDensity g γ J J s) s := (jacobi_indexPairDensity_hasDerivAt g hU hγ hJ (jacobi_smoothField g hJ) hs).const_mul 2 lemma npc_jacobiNormSqSlope_deriv_lower (g : SmoothMetric M) (hn : NonpositiveSectionalCurvature g) {γ : ℝ → M} {J : ∀ s, TangentSpace I3 (γ s)} {U : Set ℝ} (hU : IsOpen U) (hγ : ContMDiffOn 𝓘(ℝ,ℝ) I3 ∞ γ U) (hJ : IsJacobiFieldOn g γ J U) {s : ℝ} (hs : s ∈ U) : 2 * g.inner (γ s) (covariantDerivative g γ J s) (covariantDerivative g γ J s) ≤ deriv (jacobiNormSqSlope g γ J) s := by rw [(jacobiNormSqSlope_hasDerivAt g hU hγ hJ hs).deriv, indexPairDensity] have h := curvatureAlong_pairing_nonpos g J (hγ.contMDiffAt (hU.mem_nhds hs)) (nonpositiveAt_of_nonpositive g hn _) linarith lemma npc_jacobiNormSq_convex (g : SmoothMetric M) (hn : NonpositiveSectionalCurvature g) {γ : ℝ → M} {J : ∀ s, TangentSpace I3 (γ s)} {U : Set ℝ} (hU : IsOpen U) (hUc : OrdConnected U) (hγ : ContMDiffOn 𝓘(ℝ,ℝ) I3 ∞ γ U) (hJ : IsJacobiFieldOn g γ J U) : ConvexOn ℝ U (jacobiNormSq g γ J) := by apply convexOn_of_hasDerivWithinAt2_nonneg hUc.convex (intrinsic_pairing_smooth g hU hγ (jacobi_smoothField g hJ) (jacobi_smoothField g hJ)).continuousOn (f' := jacobiNormSqSlope g γ J) (f'' := fun s => 2 * indexPairDensity g γ J J s) · intro s hs exact (jacobiNormSq_hasDerivAt g hU hγ (jacobi_smoothField g hJ) (interior_subset hs)).hasDerivWithinAt · intro s hs exact (jacobiNormSqSlope_hasDerivAt g hU hγ hJ (interior_subset hs)).hasDerivWithinAt · intro s hs exact mul_nonneg (by norm_num) (indexPairDensity_nonneg_of_nonpositiveAt g J (hγ.contMDiffAt (hU.mem_nhds (interior_subset hs))) (nonpositiveAt_of_nonpositive g hn _)) end ThreeManifold.RiemannianCoordinates end section open Set Filter Topology Bundle Manifold IsManifold open scoped ContDiff namespace ThreeManifold.RiemannianCoordinates section variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] variable [CompactSpace M] [T2Space M] def fiberPhase (x : M) (v : E3) : TangentBundle I3 M := ⟨x,v⟩ lemma fiberPhase_contMDiff (x : M) : ContMDiff 𝓘(ℝ,E3) I3.tangent ∞ (fiberPhase x) := by let _ : CompactSpace M := inferInstance let _ : T2Space M := inferInstance intro v apply Bundle.contMDiffAt_totalSpace.mpr refine ⟨contMDiffAt_const,?_⟩ let e := trivializationAt E3 (TangentSpace I3) x have hmem : x ∈ e.baseSet := mem_baseSet_trivializationAt E3 (TangentSpace I3) x have he : (fun w : E3 => (e (fiberPhase x w)).2) = e.continuousLinearMapAt ℝ x := by funext w exact (e.continuousLinearMapAt_apply_of_mem ℝ hmem w).symm change ContMDiffAt 𝓘(ℝ,E3) 𝓘(ℝ,E3) ∞ (fun w => (e (fiberPhase x w)).2) v rw [he] exact (show E3 →L[ℝ] E3 from e.continuousLinearMapAt ℝ x).contDiff.contMDiff.contMDiffAt lemma fiberPhase_extChart (x : M) (v w : E3) : extChartAt I3.tangent (fiberPhase x v) (fiberPhase x w) = (chartAt E3 x x,mfderiv I3 I3 (chartAt E3 x) x w) := by let _ : CompactSpace M := inferInstance let _ : T2Space M := inferInstance apply Prod.ext · rfl · exact (congrArg (fun q : TangentBundle I3 E3 => q.2) (tangentMap_chart (I := I3) (p := fiberPhase x v) (q := fiberPhase x w) (mem_chart_source E3 x))).symm def exponential (g : SmoothMetric M) (x : M) (v : E3) : M := geodesicFamily g (fiberPhase x) v 1 lemma exponential_contMDiff (g : SmoothMetric M) (x : M) : ContMDiff 𝓘(ℝ,E3) I3 1 (exponential g x) := by intro v exact geodesicFamily_parameter_smooth g ((fiberPhase_contMDiff x).of_le (by simp)).contMDiffAt 1 lemma exponentialFamily_zero (g : SmoothMetric M) (x : M) (v : E3) : geodesicFamily g (fiberPhase x) v 0 = x := by change (geodesicFlow g (fiberPhase x v) 0).proj = x rw [geodesicFlow_zero] rfl lemma exponentialVariation_zero (g : SmoothMetric M) (x : M) (v w : E3) : geodesicVariation g (fiberPhase x) v w 0 = 0 := by unfold geodesicVariation have he : (fun y : E3 => geodesicFamily g (fiberPhase x) y 0) = fun _ => x := by funext y; exact exponentialFamily_zero g x y rw [he,mfderiv_const] rfl lemma fiberPhase_extChart_fderiv (x : M) (v b : E3) : fderiv ℝ (fun w => extChartAt I3.tangent (fiberPhase x v) (fiberPhase x w)) v b = (0,mfderiv I3 I3 (chartAt E3 x) x b) := by let L : E3 →L[ℝ] E3 := mfderiv I3 I3 (chartAt E3 x) x have hd : HasFDerivAt (fun w => (chartAt E3 x x,L w)) ((0 : E3 →L[ℝ] E3).prod L) v := (hasFDerivAt_const (chartAt E3 x x) v).prodMk L.hasFDerivAt have he : (fun w => extChartAt I3.tangent (fiberPhase x v) (fiberPhase x w)) = fun w => (chartAt E3 x x,L w) := funext (fiberPhase_extChart x v) rw [he,hd.fderiv] rfl theorem exponential_mfderiv_injective (g : SmoothMetric M) (hn : NoConjugatePoints g) (x : M) (v : E3) : Function.Injective (mfderiv 𝓘(ℝ,E3) I3 (exponential g x) v) := by suffices h : ∀ b : E3, mfderiv 𝓘(ℝ,E3) I3 (exponential g x) v b = 0 → b = 0 by intro a b hab have hz : mfderiv 𝓘(ℝ,E3) I3 (exponential g x) v (a-b) = 0 := by rw [map_sub,hab,sub_self] exact sub_eq_zero.mp (h (a-b) hz) intro b hb let f := fiberPhase x let γ := geodesicFamily g f v let J := geodesicVariation g f v b have hf : ContMDiffAt 𝓘(ℝ,E3) I3.tangent 1 f v := ((fiberPhase_contMDiff x).of_le (by simp)).contMDiffAt have hγ := geodesicFamily_isGeodesic g f v have hJ := geodesicVariation_isJacobi g hf b have hz : ∀ t : ℝ, J t = 0 := fun t => hn univ isOpen_univ ordConnected_univ γ hγ J hJ 0 (mem_univ 0) 1 (mem_univ 1) (by norm_num) (exponentialVariation_zero g x v b) hb t (mem_univ t) let c := chartAt E3 x let e := extChartAt I3.tangent (fiberPhase x v) let z := fun t => e (geodesicFlow g (f v) t) let w := fun t => fderiv ℝ (fun y => e (geodesicFlow g (f y) t)) v b have hlin := geodesic_family_linearization_at g hf 0 b simp only [geodesicFlow_zero] at hlin obtain ⟨U,hU,h0,hzt,hdz,hdw⟩ := hlin have he : (fun t => (w t).1) =ᶠ[𝓝 (0:ℝ)] fun _ => (0:E3) := by have hbase : γ 0 = x := exponentialFamily_zero g x v have hsource : γ 0 ∈ c.source := hbase ▸ mem_chart_source E3 x filter_upwards [(hγ.1.contMDiffAt (by simp)).continuousAt.preimage_mem_nhds (c.open_source.mem_nhds hsource)] with t ht have hc := geodesicVariation_coordinates g hf b t (p := fiberPhase x v) ht rw [← hc] change mfderiv I3 I3 c (γ t) (J t) = 0 rw [hz t,map_zero] have hd : HasDerivAt (fun t => (w t).1) (w 0).2 0 := by have hd' := (hdw 0 h0).congr_deriv (geodesicPhase_fderiv_apply g (chart_mem_atlas E3 x) (hzt 0 h0) (w 0)) exact hd'.fst have hw0 : (w 0).2 = 0 := by have hh := hd.congr_of_eventuallyEq he.symm exact hh.unique (hasDerivAt_const 0 (0:E3)) have hi : w 0 = (0,mfderiv I3 I3 c x b) := by dsimp only [w] simp only [geodesicFlow_zero] exact fiberPhase_extChart_fderiv x v b rw [hi] at hw0 exact (mdifferentiable_of_mem_atlas (I := I3) (chart_mem_atlas E3 x)).mfderiv_injective (mem_chart_source E3 x) (hw0.trans (map_zero _).symm) end variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] lemma isLocalHomeomorph_of_mfderiv_injective {f : E3 → M} (hf : ContMDiff 𝓘(ℝ,E3) I3 1 f) (hi : ∀ a, Function.Injective (mfderiv 𝓘(ℝ,E3) I3 f a)) : IsLocalHomeomorph f := by apply isLocalHomeomorph_iff_isOpenEmbedding_restrict.mpr intro a let c := chartAt E3 (f a) let F : E3 → E3 := c ∘ f have hc : c ∈ atlas E3 M := chart_mem_atlas E3 (f a) have hca : f a ∈ c.source := mem_chart_source E3 (f a) have hC : ContMDiffAt I3 I3 1 c (f a) := contMDiffAt_extChartAt' hca have hF : ContDiffAt ℝ 1 F a := contMDiffAt_iff_contDiffAt.mp (hC.comp a hf.contMDiffAt) have hd : fderiv ℝ F a = (mfderiv I3 I3 c (f a)).comp (mfderiv 𝓘(ℝ,E3) I3 f a) := by rw [← mfderiv_eq_fderiv] exact mfderiv_comp a (hC.mdifferentiableAt (by simp)) (hf.mdifferentiable (by simp) a) have hdi : Function.Injective (fderiv ℝ F a) := by rw [hd] exact ((mdifferentiable_of_mem_atlas (I := I3) hc).mfderiv_injective hca).comp (hi a) let L : E3 ≃L[ℝ] E3 := (LinearEquiv.ofInjectiveEndo (fderiv ℝ F a).toLinearMap hdi).toContinuousLinearEquiv have hL : (L : E3 →L[ℝ] E3) = fderiv ℝ F a := rfl have hFd : HasFDerivAt F (L : E3 →L[ℝ] E3) a := by rw [hL] exact (hF.differentiableAt (by simp)).hasFDerivAt let e := hF.toOpenPartialHomeomorph F hFd (by simp) have hea : a ∈ e.source := hF.mem_toOpenPartialHomeomorph_source hFd (by simp) let U := e.source ∩ f ⁻¹' c.source have hU : IsOpen U := e.open_source.inter (c.open_source.preimage hf.continuous) have ha : a ∈ U := ⟨hea,hca⟩ let d := (e.trans c.symm).restr U have heq : d.source = U := by simp only [d,OpenPartialHomeomorph.restr_source,OpenPartialHomeomorph.trans_source, hU.interior_eq] ext y constructor · exact fun h => h.2 · intro hy exact ⟨⟨hy.1,c.map_source hy.2⟩,hy⟩ have hequ : U.domRestrict f = d.source.domRestrict d ∘ (Homeomorph.setCongr heq.symm) := by funext y change f y.1 = c.symm (c (f y.1)) exact (c.left_inv y.2.2).symm refine ⟨U,hU.mem_nhds ha,?_⟩ rw [hequ] exact d.isOpenEmbedding_restrict.comp (Homeomorph.setCongr heq.symm).isOpenEmbedding variable [CompactSpace M] [T2Space M] theorem exponential_isLocalHomeomorph (g : SmoothMetric M) (hn : NoConjugatePoints g) (x : M) : IsLocalHomeomorph (exponential g x) := isLocalHomeomorph_of_mfderiv_injective (exponential_contMDiff g x) (exponential_mfderiv_injective g hn x) theorem npc_exponential_isLocalHomeomorph (g : SmoothMetric M) (hn : NonpositiveSectionalCurvature g) (x : M) : IsLocalHomeomorph (exponential g x) := exponential_isLocalHomeomorph g (npc_noConjugatePoints g hn) x end ThreeManifold.RiemannianCoordinates end section open Set Filter Topology Bundle Manifold IsManifold MeasureTheory open scoped ContDiff namespace ThreeManifold.RiemannianCoordinates variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] lemma metric_cauchy_schwarz (g : SmoothMetric M) (p : M) (v w : TangentSpace I3 p) : g.inner p v w ^ 2 ≤ g.inner p v v * g.inner p w w := by have h (t : ℝ) := inner_self_nonneg g p (v + t • w) simp only [map_add,map_smul,_root_.add_apply,_root_.smul_apply,smul_eq_mul] at h rw [g.symm p w v] at h have hd := discrim_le_zero (a := g.inner p w w) (b := 2*g.inner p v w) (c := g.inner p v v) (fun t => by nlinarith only [h t]) unfold discrim at hd nlinarith only [hd] def jacobiNorm (g : SmoothMetric M) (γ : ℝ → M) (J : ∀ t, TangentSpace I3 (γ t)) : ℝ → ℝ := fun t => Real.sqrt (jacobiNormSq g γ J t) def jacobiRadialPairing (g : SmoothMetric M) (γ : ℝ → M) (J : ∀ t, TangentSpace I3 (γ t)) (t : ℝ) : ℝ := g.inner (γ t) (covariantDerivative g γ J t) (J t) lemma jacobiNorm_hasDerivAt (g : SmoothMetric M) {γ : ℝ → M} {J : ∀ t, TangentSpace I3 (γ t)} {U : Set ℝ} (hU : IsOpen U) (hγ : ContMDiffOn 𝓘(ℝ,ℝ) I3 ∞ γ U) (hJ : IsJacobiFieldOn g γ J U) {s : ℝ} (hs : s ∈ U) (hJs : J s ≠ 0) : HasDerivAt (jacobiNorm g γ J) (jacobiRadialPairing g γ J s / jacobiNorm g γ J s) s := by have hp : 0 < jacobiNormSq g γ J s := g.pos _ _ hJs have hd := (jacobiNormSq_hasDerivAt g hU hγ (jacobi_smoothField g hJ) hs).sqrt (ne_of_gt hp) change HasDerivAt (fun t => Real.sqrt (jacobiNormSq g γ J t)) _ s apply hd.congr_deriv dsimp only [jacobiNormSqSlope,jacobiRadialPairing,jacobiNorm] ring lemma npc_jacobiNormSlope_hasDerivAt (g : SmoothMetric M) (hn : NonpositiveSectionalCurvature g) {γ : ℝ → M} {J : ∀ t, TangentSpace I3 (γ t)} {U : Set ℝ} (hU : IsOpen U) (hγ : ContMDiffOn 𝓘(ℝ,ℝ) I3 ∞ γ U) (hJ : IsJacobiFieldOn g γ J U) {s : ℝ} (hs : s ∈ U) (hJs : J s ≠ 0) : ∃ a : ℝ, 0 ≤ a ∧ HasDerivAt (fun t => jacobiRadialPairing g γ J t / jacobiNorm g γ J t) a s := by let A := jacobiRadialPairing g γ J s let Q := jacobiNormSq g γ J s let N := jacobiNorm g γ J s let K := indexPairDensity g γ J J s have hQ : 0 < Q := g.pos _ _ hJs have hN : 0 < N := Real.sqrt_pos.mpr hQ have hN2 : N^2 = Q := Real.sq_sqrt hQ.le have hdA : HasDerivAt (jacobiRadialPairing g γ J) K s := jacobi_indexPairDensity_hasDerivAt g hU hγ hJ (jacobi_smoothField g hJ) hs have hdN := jacobiNorm_hasDerivAt g hU hγ hJ hs hJs refine ⟨(K*N-A*(A/N))/N^2,?_,hdA.div hdN (ne_of_gt hN)⟩ apply div_nonneg _ (sq_nonneg N) have hcs := metric_cauchy_schwarz g (γ s) (covariantDerivative g γ J s) (J s) have hc := curvatureAlong_pairing_nonpos g J (hγ.contMDiffAt (hU.mem_nhds hs)) (nonpositiveAt_of_nonpositive g hn _) have hK : g.inner (γ s) (covariantDerivative g γ J s) (covariantDerivative g γ J s) ≤ K := by dsimp [K,indexPairDensity] linarith only [hc] have hmul := mul_le_mul_of_nonneg_right hK hQ.le have hmain : A^2 ≤ K*Q := hcs.trans hmul have he : (K*N-A*(A/N))*N = K*Q-A^2 := by rw [← hN2] field_simp exact (mul_nonneg_iff_of_pos_right hN).mp (by rw [he]; linarith only [hmain]) theorem npc_jacobiNorm_convex_Ici (g : SmoothMetric M) (hn : NonpositiveSectionalCurvature g) {γ : ℝ → M} {J : ∀ t, TangentSpace I3 (γ t)} (hγ : ContMDiff 𝓘(ℝ,ℝ) I3 ∞ γ) (hJ : IsJacobiFieldOn g γ J univ) (hpos : ∀ t > 0, J t ≠ 0) : ConvexOn ℝ (Ici 0) (jacobiNorm g γ J) := by classical have hc : Continuous (jacobiNorm g γ J) := by apply Real.continuous_sqrt.comp apply continuousOn_univ.mp exact (intrinsic_pairing_smooth g isOpen_univ hγ.contMDiffOn (jacobi_smoothField g hJ) (jacobi_smoothField g hJ)).continuousOn have hd (s : ℝ) (hs : s ∈ interior (Ici (0:ℝ))) := npc_jacobiNormSlope_hasDerivAt g hn isOpen_univ hγ.contMDiffOn hJ (mem_univ s) (hpos s (by simpa only [interior_Ici,mem_Ioi] using hs)) choose a ha hda using hd apply convexOn_of_hasDerivWithinAt2_nonneg (convex_Ici 0) hc.continuousOn (f' := fun t => jacobiRadialPairing g γ J t / jacobiNorm g γ J t) (f'' := fun s => if hs : s ∈ interior (Ici (0:ℝ)) then a s hs else 0) · intro s hs exact (jacobiNorm_hasDerivAt g isOpen_univ hγ.contMDiffOn hJ (mem_univ s) (hpos s (by simpa only [interior_Ici,mem_Ioi] using hs))).hasDerivWithinAt · intro s hs simpa only [dite_eq_left hs] using (hda s hs).hasDerivWithinAt (s := interior (Ici (0:ℝ))) · intro s hs simpa only [dite_eq_left hs] using ha s hs lemma connectionTerm_zero_right (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3) (x v : E3) : connectionTerm g c x v 0 = 0 := by ext i simp only [connectionTerm,Pi.zero_apply,mul_zero,Finset.sum_const_zero] lemma jacobiNorm_initial_rightDerivative (g : SmoothMetric M) {γ : ℝ → M} {J : ∀ t, TangentSpace I3 (γ t)} (hγ : ContMDiff 𝓘(ℝ,ℝ) I3 ∞ γ) (hJ : SmoothFieldOn γ J univ) (h0 : J 0 = 0) : HasDerivWithinAt (jacobiNorm g γ J) (Real.sqrt (g.inner (γ 0) (covariantDerivative g γ J 0) (covariantDerivative g γ J 0))) (Ioi 0) 0 := by let c := chartAt E3 (γ 0) let j := fieldCoordinates γ J c let v := deriv j 0 have hc : c ∈ atlas E3 M := chart_mem_atlas E3 (γ 0) have hsc : γ 0 ∈ c.source := mem_chart_source E3 (γ 0) have hjs : ContDiffAt ℝ ∞ j 0 := smoothField_coordinatesAt isOpen_univ hγ.contMDiffOn hJ hc (mem_univ 0) hsc have hz : j 0 = 0 := by simp only [j,fieldCoordinates,h0,map_zero]; rfl have hdj : HasDerivAt j v 0 := (hjs.differentiableAt (by simp)).hasDerivAt have hlimj : Tendsto (fun t : ℝ => t⁻¹ • j t) (𝓝[>] 0) (𝓝 v) := by simpa only [zero_add,hz,sub_zero] using hdj.tendsto_slope_zero_right have hcov : fieldCoordinates γ (covariantDerivative g γ J) c 0 = v := by rw [covariantDerivative_coordinates g hγ.contMDiffAt (hjs.differentiableAt (by simp)) hc hsc] change deriv j 0 + connectionTerm g c (c (γ 0)) (deriv (c ∘ γ) 0) (j 0) = v rw [hz,connectionTerm_zero_right,add_zero] have hB : ContinuousAt (fun t : ℝ => chartBilin g c (c (γ t))) 0 := by change ContinuousAt (chartBilin g c ∘ (c ∘ γ)) 0 exact ContinuousAt.comp (f := c ∘ γ) (chartBilin_smoothAt g hc (c.map_source hsc)).continuousAt ((c.continuousAt hsc).comp hγ.continuous.continuousAt) have hlimB := hB.tendsto.mono_left (show 𝓝[>] (0:ℝ) ≤ 𝓝 0 from nhdsWithin_le_nhds) have hcont₁ : Continuous (fun p : (E3 →L[ℝ] E3 →L[ℝ] ℝ) × E3 => p.1 p.2) := continuous_fst.clm_apply continuous_snd have hlim₁ := hcont₁.tendsto (chartBilin g c (c (γ 0)),v) |>.comp (hlimB.prodMk_nhds hlimj) have hcont₂ : Continuous (fun p : (E3 →L[ℝ] ℝ) × E3 => p.1 p.2) := continuous_fst.clm_apply continuous_snd have hlim₂ := hcont₂.tendsto (chartBilin g c (c (γ 0)) v,v) |>.comp (hlim₁.prodMk_nhds hlimj) have hlim := Real.continuous_sqrt.tendsto (chartBilin g c (c (γ 0)) v v) |>.comp hlim₂ have hbase : chartBilin g c (c (γ 0)) v v = g.inner (γ 0) (covariantDerivative g γ J 0) (covariantDerivative g γ J 0) := by rw [← hcov] exact metric_field_coordinates g _ _ hc hsc rw [hbase] at hlim apply (hasDerivWithinAt_iff_tendsto_slope' self_notMem_Ioi).mpr apply hlim.congr' have hnorm0 : jacobiNorm g γ J 0 = 0 := by simp only [jacobiNorm,jacobiNormSq,h0,map_zero,Real.sqrt_zero] have hsource : γ ⁻¹' c.source ∈ 𝓝[>] (0:ℝ) := nhdsWithin_le_nhds (hγ.continuous.continuousAt.preimage_mem_nhds (c.open_source.mem_nhds hsc)) filter_upwards [self_mem_nhdsWithin,hsource] with t ht htc change Real.sqrt (chartBilin g c (c (γ t)) (t⁻¹ • j t) (t⁻¹ • j t)) = slope (jacobiNorm g γ J) 0 t have hm : chartBilin g c (c (γ t)) (j t) (j t) = jacobiNormSq g γ J t := metric_field_coordinates g J J hc htc simp only [map_smul,_root_.smul_apply,smul_eq_mul] rw [← mul_assoc,← sq,hm,Real.sqrt_mul (sq_nonneg _),Real.sqrt_sq (inv_nonneg.mpr ht.le)] rw [slope_def_field,hnorm0,sub_zero,sub_zero] simp only [jacobiNorm,div_eq_mul_inv,mul_comm] theorem npc_jacobiNorm_lower (g : SmoothMetric M) (hn : NonpositiveSectionalCurvature g) {γ : ℝ → M} {J : ∀ t, TangentSpace I3 (γ t)} (hγ : IsGeodesicOn g γ univ) (hJ : IsJacobiFieldOn g γ J univ) (h0 : J 0 = 0) {t : ℝ} (ht : 0 ≤ t) : t * Real.sqrt (g.inner (γ 0) (covariantDerivative g γ J 0) (covariantDerivative g γ J 0)) ≤ jacobiNorm g γ J t := by by_cases hd : covariantDerivative g γ J 0 = 0 · simp only [hd,map_zero,Real.sqrt_zero,mul_zero] exact Real.sqrt_nonneg _ have hpos : ∀ s > (0:ℝ), J s ≠ 0 := by intro s hs hsz have hzero : ∀ z : ℝ, J z = 0 := fun z => npc_noConjugatePoints g hn univ isOpen_univ ordConnected_univ γ hγ J hJ 0 (mem_univ 0) s (mem_univ s) (ne_of_lt hs) h0 hsz z (mem_univ z) apply hd dsimp only [covariantDerivative,alongDerivative] have hz : fieldCoordinates γ J (chartAt E3 (γ 0)) = fun _ => 0 := by funext r; simp only [fieldCoordinates,hzero r,map_zero]; rfl rw [hz,deriv_const,connectionTerm_zero_right,add_zero] exact map_zero _ have hγ' : ContMDiff 𝓘(ℝ,ℝ) I3 ∞ γ := contMDiffOn_univ.mp hγ.1 have hc := npc_jacobiNorm_convex_Ici g hn hγ' hJ hpos have hnorm0 : jacobiNorm g γ J 0 = 0 := by simp only [jacobiNorm,jacobiNormSq,h0,map_zero,Real.sqrt_zero] rcases ht.eq_or_lt with rfl | ht · simpa only [zero_mul,hnorm0] using (le_refl (0:ℝ)) have he := hc.le_slope_of_hasDerivWithinAt_Ioi (show (0:ℝ) ∈ Ici 0 by simp) ht.le ht (jacobiNorm_initial_rightDerivative g hγ' (jacobi_smoothField g hJ) h0) simp only [slope_def_field,hnorm0,sub_zero] at he simpa only [mul_comm] using (le_div_iff₀ ht).mp he end ThreeManifold.RiemannianCoordinates end section open Set Filter Topology Bundle Manifold IsManifold open scoped ContDiff namespace ThreeManifold.RiemannianCoordinates section variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] variable [CompactSpace M] [T2Space M] lemma exponentialVariation_initial_covariantDerivative (g : SmoothMetric M) (x : M) (v b : E3) : covariantDerivative g (geodesicFamily g (fiberPhase x) v) (geodesicVariation g (fiberPhase x) v b) 0 = b := by let f := fiberPhase x let γ := geodesicFamily g f v let J := geodesicVariation g f v b let c := chartAt E3 x let e := extChartAt I3.tangent (fiberPhase x v) let z := fun t => e (geodesicFlow g (f v) t) let w := fun t => fderiv ℝ (fun y => e (geodesicFlow g (f y) t)) v b have hf : ContMDiffAt 𝓘(ℝ,E3) I3.tangent 1 f v := ((fiberPhase_contMDiff x).of_le (by simp)).contMDiffAt have hγ := geodesicFamily_isGeodesic g f v have hJ := geodesicVariation_isJacobi g hf b have hbase : γ 0 = x := exponentialFamily_zero g x v have hsource : γ 0 ∈ c.source := hbase ▸ mem_chart_source E3 x have hlin := geodesic_family_linearization_at g hf 0 b simp only [geodesicFlow_zero] at hlin obtain ⟨U,hU,h0,hzt,hdz,hdw⟩ := hlin have he : fieldCoordinates γ J c =ᶠ[𝓝 (0:ℝ)] fun t => (w t).1 := by filter_upwards [(hγ.1.contMDiffAt (by simp)).continuousAt.preimage_mem_nhds (c.open_source.mem_nhds hsource)] with t ht exact geodesicVariation_coordinates g hf b t (p := fiberPhase x v) ht have hd : HasDerivAt (fieldCoordinates γ J c) (w 0).2 0 := by have hd' := (hdw 0 h0).congr_deriv (geodesicPhase_fderiv_apply g (chart_mem_atlas E3 x) (hzt 0 h0) (w 0)) have hh : HasDerivAt (fun t => (w t).1) (w 0).2 0 := hd'.fst exact hh.congr_of_eventuallyEq he have hi : w 0 = (0,mfderiv I3 I3 c x b) := by dsimp only [w] simp only [geodesicFlow_zero] exact fiberPhase_extChart_fderiv x v b have hcov := covariantDerivative_coordinates g (hγ.1.contMDiffAt (by simp)) ((smoothField_coordinatesAt isOpen_univ hγ.1 (jacobi_smoothField g hJ) (chart_mem_atlas E3 (γ 0)) (mem_univ 0) (mem_chart_source E3 (γ 0))).differentiableAt (by simp)) (chart_mem_atlas E3 x) hsource have hz : fieldCoordinates γ J c 0 = 0 := by change mfderiv I3 I3 c (γ 0) (J 0) = 0 rw [show J 0 = 0 from exponentialVariation_zero g x v b,map_zero] change mfderiv I3 I3 c (γ 0) (covariantDerivative g γ J 0) = _ at hcov rw [alongDerivative,hd.deriv,hi,hz,connectionTerm_zero_right,add_zero,hbase] at hcov exact (mdifferentiable_of_mem_atlas (I := I3) (chart_mem_atlas E3 x)).mfderiv_injective (mem_chart_source E3 x) hcov theorem npc_exponential_differential_expanding (g : SmoothMetric M) (hn : NonpositiveSectionalCurvature g) (x : M) (v b : E3) : g.inner x b b ≤ g.inner (exponential g x v) (mfderiv 𝓘(ℝ,E3) I3 (exponential g x) v b) (mfderiv 𝓘(ℝ,E3) I3 (exponential g x) v b) := by have hf : ContMDiffAt 𝓘(ℝ,E3) I3.tangent 1 (fiberPhase x) v := ((fiberPhase_contMDiff x).of_le (by simp)).contMDiffAt have h := npc_jacobiNorm_lower g hn (geodesicFamily_isGeodesic g (fiberPhase x) v) (geodesicVariation_isJacobi g hf b) (exponentialVariation_zero g x v b) (t := 1) (by norm_num) rw [exponentialVariation_initial_covariantDerivative,exponentialFamily_zero,one_mul] at h change Real.sqrt (g.inner x b b) ≤ Real.sqrt (g.inner (exponential g x v) (mfderiv 𝓘(ℝ,E3) I3 (exponential g x) v b) (mfderiv 𝓘(ℝ,E3) I3 (exponential g x) v b)) at h exact (Real.sqrt_le_sqrt_iff (inner_self_nonneg g _ _)).mp h end variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] lemma deriv_comp_mul {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {k t : ℝ} (hf : DifferentiableAt ℝ f (k*t)) : deriv (fun r => f (k*r)) t = k • deriv f (k*t) := by simpa only [Function.comp_def,mul_one] using (hf.hasDerivAt.scomp t ((hasDerivAt_id t).const_mul k)).deriv lemma geodesic_scale (g : SmoothMetric M) {γ : ℝ → M} {U : Set ℝ} (hU : IsOpen U) (hγ : IsGeodesicOn g γ U) (k : ℝ) : IsGeodesicOn g (fun r => γ (k*r)) ((fun r => k*r) ⁻¹' U) := by have htA : ContMDiff 𝓘(ℝ,ℝ) 𝓘(ℝ,ℝ) ∞ (fun r : ℝ => k*r) := contMDiff_iff_contDiff.mpr (contDiff_const.mul contDiff_id) refine ⟨hγ.1.comp htA.contMDiffOn (fun _ h => h),?_⟩ intro c hc t ht hct have hs := chartPhase_contDiffAt (hγ.1.contMDiffAt (hU.mem_nhds ht)) hc hct have hn : ∀ᶠ r in 𝓝 t, k*r ∈ U ∧ γ (k*r) ∈ c.source := continuousAt_const.mul continuousAt_id |>.preimage_mem_nhds (Filter.Eventually.and (hU.mem_nhds ht) ((hγ.1.contMDiffAt (hU.mem_nhds ht)).continuousAt.preimage_mem_nhds (c.open_source.mem_nhds hct))) have hd : deriv (c ∘ (fun r => γ (k*r))) =ᶠ[𝓝 t] fun r => k • deriv (c ∘ γ) (k*r) := by filter_upwards [hn] with r hr exact deriv_comp_mul ((chartPhase_contDiffAt (hγ.1.contMDiffAt (hU.mem_nhds hr.1)) hc hr.2).fst.differentiableAt (by simp)) have hdd : deriv (deriv (c ∘ (fun r => γ (k*r)))) t = (k*k) • deriv (deriv (c ∘ γ)) (k*t) := by rw [hd.deriv_eq] change deriv (k • (fun r => deriv (c ∘ γ) (k*r))) t = _ have hdγ : DifferentiableAt ℝ (deriv (c ∘ γ)) (k*t) := hs.snd.differentiableAt (by simp) rw [deriv_const_smul k,deriv_comp_mul hdγ,smul_smul] exact hdγ.comp t (((hasDerivAt_id t).const_mul k).differentiableAt) change _ + _ = 0 rw [hdd,hd.eq_of_nhds] have hbil (u v : E3) : connectionTerm g c (c (γ (k*t))) (k • u) (k • v) = (k*k) • connectionTerm g c (c (γ (k*t))) u v := by simp only [← connectionBilin_apply,map_smul,_root_.smul_apply,smul_smul] simp only [Function.comp_apply] at * have hgeo := hγ.2 c hc (k*t) ht hct change deriv (deriv (c ∘ γ)) (k*t) + connectionTerm g c (c (γ (k*t))) (deriv (c ∘ γ) (k*t)) (deriv (c ∘ γ) (k*t)) = 0 at hgeo rw [hbil,← smul_add,hgeo,smul_zero] lemma geodesic_const (g : SmoothMetric M) (x : M) : IsGeodesicOn g (fun _ : ℝ => x) univ := by refine ⟨contMDiff_const.contMDiffOn,?_⟩ intro c hc t ht hxt simp only [Function.comp_def,alongDerivative,deriv_const] simp only [← connectionBilin_apply,map_zero] simp [deriv_const] variable [CompactSpace M] [T2Space M] lemma exponential_zero (g : SmoothMetric M) (x : M) : exponential g x 0 = x := by let γ := globalGeodesic g (fiberPhase x 0) have hγ := globalGeodesic_isGeodesic g (fiberPhase x 0) have he : phaseLift γ 0 = phaseLift (fun _ : ℝ => x) 0 := by rw [globalGeodesic_initial] unfold fiberPhase phaseLift velocity rw [mfderiv_const] rfl exact geodesic_unique g isOpen_univ ordConnected_univ hγ (geodesic_const g x) (mem_univ 0) he (mem_univ 1) lemma exponential_scale (g : SmoothMetric M) (x : M) (v : E3) (k : ℝ) : exponential g x (k • v) = globalGeodesic g (fiberPhase x v) k := by let γ := globalGeodesic g (fiberPhase x v) have hγ := globalGeodesic_isGeodesic g (fiberPhase x v) have hk : IsGeodesicOn g (fun r => γ (k*r)) univ := by simpa only [preimage_univ] using geodesic_scale g isOpen_univ hγ k have he : phaseLift (globalGeodesic g (fiberPhase x (k • v))) 0 = phaseLift (fun r => γ (k*r)) 0 := by rw [globalGeodesic_initial] have hbase : γ (k*0) = x := by change globalGeodesic g (fiberPhase x v) (k*0) = x rw [mul_zero] exact congrArg Bundle.TotalSpace.proj (globalGeodesic_initial g (fiberPhase x v)) apply Bundle.TotalSpace.ext hbase.symm apply heq_of_eq change k • v = mfderiv 𝓘(ℝ,ℝ) I3 (fun r => γ (k*r)) 0 1 have hc := mfderiv_comp 0 ((hγ.1.contMDiffAt (by simp)).mdifferentiableAt (by simp)) ((contMDiff_iff_contDiff.mpr (contDiff_const.mul contDiff_id) : ContMDiff 𝓘(ℝ,ℝ) 𝓘(ℝ,ℝ) ∞ (fun r : ℝ => k*r)).mdifferentiable (by simp) 0) have hmul : mfderiv 𝓘(ℝ,ℝ) 𝓘(ℝ,ℝ) (fun r : ℝ => k*r) 0 1 = k := by rw [mfderiv_eq_fderiv] change deriv (fun r : ℝ => k*r) 0 = k simpa only [mul_one,id_eq] using ((hasDerivAt_id (0:ℝ)).const_mul k).deriv have hv : (velocity γ 0 : E3) = v := congrArg (fun p : TangentBundle I3 M => (p.snd : E3)) (globalGeodesic_initial g (fiberPhase x v)) have hh := congrArg (fun L : ℝ →L[ℝ] E3 => L 1) hc change mfderiv 𝓘(ℝ,ℝ) I3 (fun r => γ (k*r)) 0 1 = mfderiv 𝓘(ℝ,ℝ) I3 γ (k*0) (mfderiv 𝓘(ℝ,ℝ) 𝓘(ℝ,ℝ) (fun r : ℝ => k*r) 0 1) at hh rw [hmul,mul_zero] at hh have hsm : (mfderiv 𝓘(ℝ,ℝ) I3 γ 0 k : E3) = k • (velocity γ 0 : E3) := by unfold velocity let L : ℝ →L[ℝ] E3 := mfderiv 𝓘(ℝ,ℝ) I3 γ 0 change L k = k • L 1 have hL := L.map_smul k (1:ℝ) simp only [smul_eq_mul,mul_one] at hL exact hL exact ((congrArg (fun w : E3 => k • w) hv).symm.trans hsm.symm).trans hh.symm have hh := geodesic_unique g isOpen_univ ordConnected_univ (globalGeodesic_isGeodesic g (fiberPhase x (k • v))) hk (mem_univ 0) he (mem_univ 1) change globalGeodesic g (fiberPhase x (k • v)) 1 = globalGeodesic g (fiberPhase x v) k simpa only [mul_one] using hh end ThreeManifold.RiemannianCoordinates end open Set Filter Topology Bundle Manifold IsManifold open scoped ContDiff namespace ThreeManifold.RiemannianCoordinates variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M] [IsManifold I3 ∞ M] lemma curvatureAlong_velocity_pairing (g : SmoothMetric M) {γ : ℝ → M} (J : ∀ s, TangentSpace I3 (γ s)) {s : ℝ} (hγ : ContMDiffAt 𝓘(ℝ,ℝ) I3 ∞ γ s) : g.inner (γ s) (curvatureAlong g γ J s) (velocity γ s) = 0 := by let c := chartAt E3 (γ s) have hc := chart_mem_atlas E3 (γ s) have hx := mem_chart_source E3 (γ s) rw [← metric_field_coordinates g (curvatureAlong g γ J) (velocity γ) hc hx, curvatureAlong_coordinates g J hγ hc hx,velocity_coordinates hγ hc hx] have h := curvature_pairing_skew g hc (c.map_source hx) (fieldCoordinates γ J c s) (deriv (c ∘ γ) s) (deriv (c ∘ γ) s) (deriv (c ∘ γ) s) rw [chartBilin_symm g c] at h rw [chartBilin_symm g (chartAt E3 (γ s))] change _ + _ = 0 at h linarith lemma jacobi_velocity_pairing (g : SmoothMetric M) {γ : ℝ → M} (hγ : IsGeodesicOn g γ univ) {J : ∀ s, TangentSpace I3 (γ s)} (hJ : IsJacobiFieldOn g γ J univ) (h0 : J 0 = 0) (s : ℝ) : g.inner (γ s) (J s) (velocity γ s) = s * g.inner (γ 0) (covariantDerivative g γ J 0) (velocity γ 0) := by let D := covariantDerivative g γ J have hD := covariantDerivative_smooth g isOpen_univ hγ.1 (jacobi_smoothField g hJ) have hd (t : ℝ) : HasDerivAt (fun r => g.inner (γ r) (D r) (velocity γ r)) 0 t := by have h := intrinsic_pairing_hasDerivAt g isOpen_univ hγ.1 hD (velocity_smooth isOpen_univ hγ.1) (mem_univ t) have he := jacobi_intrinsic_equation g isOpen_univ hγ.1 hJ (mem_univ t) change covariantDerivative g γ D t + curvatureAlong g γ J t = 0 at he have he' := eq_neg_of_add_eq_zero_left he dsimp only [D] at he' simpa only [he',geodesic_acceleration_zero g isOpen_univ hγ (mem_univ t),map_zero, map_neg,_root_.neg_apply,curvatureAlong_velocity_pairing g J (hγ.1.contMDiffAt (by simp)), neg_zero,add_zero] using h have hconst (t : ℝ) : g.inner (γ t) (D t) (velocity γ t) = g.inner (γ 0) (D 0) (velocity γ 0) := isOpen_univ.is_const_of_deriv_eq_zero (convex_univ : Convex ℝ (univ : Set ℝ)).isPreconnected (fun t _ => (hd t).differentiableAt.differentiableWithinAt) (fun t _ => (hd t).deriv) (mem_univ t) (mem_univ 0) have hp (t : ℝ) : HasDerivAt (fun r => g.inner (γ r) (J r) (velocity γ r)) (g.inner (γ 0) (D 0) (velocity γ 0)) t := by have h := intrinsic_pairing_hasDerivAt g isOpen_univ hγ.1 (jacobi_smoothField g hJ) (velocity_smooth isOpen_univ hγ.1) (mem_univ t) simp only [geodesic_acceleration_zero g isOpen_univ hγ (mem_univ t),map_zero,add_zero] at h exact h.congr_deriv (hconst t) have hh := isOpen_univ.eqOn_of_deriv_eq (convex_univ : Convex ℝ (univ : Set ℝ)).isPreconnected (fun t _ => (hp t).differentiableAt.differentiableWithinAt) ((differentiable_id.mul_const (g.inner (γ 0) (D 0) (velocity γ 0))).differentiableOn) (fun t _ => by rw [(hp t).deriv]; simp) (mem_univ 0) (by simp only [h0,map_zero,_root_.zero_apply,zero_mul,id_eq]) exact hh (mem_univ s) variable [CompactSpace M] [T2Space M] lemma exponential_radial_differential (g : SmoothMetric M) (p : M) (v : E3) : mfderiv 𝓘(ℝ,E3) I3 (exponential g p) v v = velocity (globalGeodesic g (fiberPhase p v)) 1 := by have he : (fun t : ℝ => exponential g p (t • v)) = globalGeodesic g (fiberPhase p v) := funext (exponential_scale g p v) have hh := mfderiv_comp (I := 𝓘(ℝ,ℝ)) (I' := 𝓘(ℝ,E3)) (I'' := I3) (f := fun t : ℝ => t • v) (g := exponential g p) (1:ℝ) ((exponential_contMDiff g p).mdifferentiable (by simp) (1 • v)) ((contDiff_id.smul contDiff_const : ContDiff ℝ 1 (fun t : ℝ => t • v)).differentiable (by simp) 1).mdifferentiableAt have h := congrArg (fun L : ℝ →L[ℝ] E3 => L 1) hh simp only [Function.comp_def] at h rw [he] at h have hd : mfderiv 𝓘(ℝ,ℝ) 𝓘(ℝ,E3) (fun t : ℝ => t • v) 1 1 = v := by rw [mfderiv_eq_fderiv] change deriv (fun t : ℝ => t • v) 1 = v simpa using ((hasDerivAt_id (1:ℝ)).smul_const v).deriv change velocity (globalGeodesic g (fiberPhase p v)) 1 = mfderiv 𝓘(ℝ,E3) I3 (exponential g p) (1 • v) (mfderiv 𝓘(ℝ,ℝ) 𝓘(ℝ,E3) (fun t : ℝ => t • v) 1 1) at h rw [hd,one_smul] at h exact h.symm theorem exponential_gauss (g : SmoothMetric M) (p : M) (v b : E3) : g.inner (exponential g p v) (mfderiv 𝓘(ℝ,E3) I3 (exponential g p) v b) (mfderiv 𝓘(ℝ,E3) I3 (exponential g p) v v) = g.inner p b v := by have hf := ((fiberPhase_contMDiff p).of_le (show (1 : WithTop ℕ∞) ≤ ∞ by simp)).contMDiffAt (x := v) have hh := jacobi_velocity_pairing g (geodesicFamily_isGeodesic g (fiberPhase p) v) (geodesicVariation_isJacobi g hf b) (exponentialVariation_zero g p v b) 1 rw [exponentialVariation_initial_covariantDerivative,exponentialFamily_zero,one_mul] at hh have hv : (velocity (geodesicFamily g (fiberPhase p) v) 0 : E3) = v := congrArg (fun q : TangentBundle I3 M => (q.snd : E3)) (globalGeodesic_initial g (fiberPhase p v)) rw [hv] at hh rw [exponential_radial_differential] exact hh end ThreeManifold.RiemannianCoordinates end end OAI