Cut loci and diameters of the Berger lens spaces

Cut lo ci and diameters of the Berger lens spaces ∗ A. V. P o dobry aev A. K. Ailamazy an Program Systems Institute of RAS alex@alex.botik.ru Abstract In this pap er, we study Riemannian metrics on the three-dimensional lens spaces that are deformations of the standard Riemannian metric along the fib ers of the Hopf fibration. In other w ords, these metrics are axisymmetric. There is a one-parametric family of suc h metrics. This family tends to an axisymmetric sub-Riemannian metric. W e find the cut lo ci and the cut times using methods from geometric control theory . It turns out that the cut lo ci and the cut times con verge to the cut lo cus and the cut time for the sub-Riemannian structure, that was already studied. Moreo v er, we get some lo w er bounds for the diameter of these Riemannian metrics. These b ounds coincide with the exact v alues of diameters for the lens spaces L ( p ; 1). Keyw ords : Lens space, Berger sphere, cut lo cus, diameter, geometric control theory . AMS sub ject classification : 53C30, 53C17, 49J15. 1 In tro duction This pap er is a sequel of pap ers [ 1 , 2 ] where we found cut lo ci and diameters of left-inv ariant axisym- metric Riemannian metrics on the Lie groups SU 2 and SO 3 . Note that the three-dimensional sphere SU 2 with suc h a metric is a Berger sphere, i.e., a sphere with the standard metric deformed along the fib ers of the Hopf fibration. A t the same time, the Lie group SO 3 = SU 2 / Z 2 is a sp ecial case of a lens space. Therefore, it seems natural to study the cut lo ci of general lens spaces with this kind of metric. Moreo v er, our previous works [ 1 , 3 , 4 ] w ere motiv ated by the fact that axisymmetric left-inv ariant sub-Riemannian structures on the Lie groups SL 2 , SO 3 , and SU 2 w ere studied in the w orks of V. N. Be- resto vskii and I. A. Zubarev a [ 5 , 6 , 7 ] and U. Boscain and F. Rossi [ 8 ]. W e inv estigated one-parameter families of Riemannian metrics tending to these sub-Riemannian structures. In pap er [ 8 ], the authors also considered sub-Riemannian structures on lens spaces. In the presen t pap er, w e consider one- parameter families of in v arian t axisymmetric Riemannian metrics on lens spaces whose limits are the sub-Riemannian metrics studied in [ 8 ]. Moreo v er, considering the family of the lens spaces L ( p ; q ) dep ending on the parameter p allows us to b etter understand the reasons for the app earance of an additional strata of the cut lo cus that arise in the cases of SU 2 and SO 3 . Notice that lens spaces L ( p ; q ) are not homogeneous spaces. So, the cut lo cus dep ends on an initial p oin t of the lens space. W e compute the cut lo cus with resp ect to the initial p oin t o = Π(id), where id ∈ SU 2 is the identit y element and Π : SU 2 ≃ S 3 → L ( p ; q ) is the natural pro jection. Ho wev er, the lens spaces L ( p ; 1) and L ( p ; − 1) are homogeneous with resp ect to the SU 2 -action. In these cases our results don’t dep end on an initial p oint. It should b e noted that the cut lo cus and the diameter of the standard Riemannian metric on lens spaces are kno wn from the results of S. Anisov [ 9 ]. The present pap er generalizes these results to the case of metrics deformed along the fib ers of the Hopf fibration, which w e call the Berger metrics. The pap er has the follo wing structure. In Section 2 we recall the definition of the lens space and define a Riemannian metric of the Berger t yp e. Then, in Section 3 w e give several necessary definitions ∗ This w ork w as supp orted b y the Russian Science F oundation under grant no. 25-21-00681, https://rscf.ru/pro ject/25- 21-00681/ and p erformed in Ailamazy an Program Systems Institute of Russian Academy of Sciences. 1 Figure 1: The lens space L ( p ; q ), p ⩾ 2 as a domain L ⊂ R 3 with iden tified p oints on its b oundary ∂ L . Iden tified p oints ha v e the same marks. The general view (left) and the pro jection to the ( q 1 , q 2 )-plane (righ t) for p = 8 and q = 3. and some previous results on geo desics equations and the first conjugate time. Section 4 is dedicated to the relative lo cation of Maxwell strata. Then, w e pro ve that the first Maxw ell time for symmetries is not greater than the first conjugate time in Section 5 . Section 6 con tains the main result (Theorem 1 ) whic h describ es the cut lo cus, also there is Theorem 2 ab out the sub-Riemannian structure as a limit case of Riemannian ones. Finally , w e find lo w er b ounds for diameters of the Berger lens spaces, see Theorem 3 in Section 7 . 2 Axisymmetric Riemannian metrics on lens spaces Let us recall the notion of a lens space. Definition 1. Let p, q ∈ Z \ { 0 } b e coprime num b ers. Consider a three dimensional sphere S 3 = { ( z , w ) ∈ C 2 | | z | 2 + | w | 2 = 1 } and a Z p -action on S 3 via the formula: [ k ] ◦ ( z , w ) =  e i 2 πk p z , e i 2 πk q p w  , where [ k ] ∈ Z p , ( z , w ) ∈ S 3 . The orbit space L ( p ; q ) = S 3 / Z p is called a lens sp ac e . W e denote by Π : S 3 → L ( p ; q ) the natural pro jection. W e need some mo del of the lens space L ( p ; q ). Let z = q 0 + iq 3 , w = q 1 + iq 2 . Prop osition 1 (See, for example, [ 8 ], Prop. 2) . Assume that p > 1 . Consider the subset L = ( ( z , w ) ∈ S 3    q 2 1 + q 2 2 + q 2 3 sin 2 π p ⩽ 1 , q 0 ⩾ 0 ) ⊂ S 3 . L et us glue the p oints ( q 1 , q 2 , q 3 ) ∼ ( q ′ 1 , q ′ 2 , q ′ 3 ) of the b oundary ∂ L if one of the fol lowing c onditions holds : (1) q 3 > 0 , q ′ 3 = − q 3 and z ′ = q ′ 1 + iq ′ 2 = e 2 πq p z , wher e z = q 1 + iq 2 , (2) q 3 = q ′ 3 = 0 , and z ′ = e 2 πk p z for some k = 1 , . . . , p . The r esulting top olo gic al sp ac e is home omorphic to L ( p ; q ) , se e Fig. 1 . Remark 1. It is not difficult to see that there are homeomorphisms L ( p ; q ) ≃ L ( − p ; q ) ≃ L ( p ; − q ) ≃ L ( p ; k p + q ) for k ∈ Z . Let us mention some particular cases. First, note that the Lie group SU 2 =  z w − ¯ w ¯ z     z , w ∈ C , | z | 2 + | w | 2 = 1  2 is homeomorphic to S 3 . Second, the lens spaces L ( p ; 1) are SU 2 -homogeneous. Indeed , the action of the generator of the group Z p can b e represen ted by the left multiplication b y the matrix e i 2 π p 0 0 e − i 2 π p ! whic h comm utes with the right action of the group SU 2 . Similarly , the lens space L ( p ; − 1) is SU 2 - homogeneous, since the Z p -generator’s action can b e represented b y the righ t multiplication b y the same matrix that comm utes with the left action of the group SU 2 . Ob viously , L (1; 1) = S 3 = SU 2 and L (2; 1) = R P 2 = SO 3 . The lens space L (4; 1) is homeomorphic to the pro jectivization of the tangent bundle of the tw o dimensional sphere. It may b e a useful mo del for an anthropomorphic con tour reco v ery for spherical images a voiding cusps, compare with [ 10 , 11 ]. No w we define a Riemannian metric we deal with. Notice that the left action of the circle group U 1 =  e iφ 0 0 e − iφ     φ ∈ R  ⊂ SU 2 on SU 2 comm utes with the Z p -action. This implies, that the group U 1 acts also on the lens space L ( p ; q ). The factor is homeomorphic to S 2 , since it is a complex pro jectiv e line C P 1 with homogeneous co ordinates [ z : w ] factorized by cyclic group generated by rotation by the angle 2 π (1 − q ) p around zero in the affine chart w  = 0. Definition 2. The natural pro jection L ( p ; q ) → L ( p ; q ) / U 1 ≃ S 2 is called the Hopf fibr ation . Its fib ers are circles. W e consider the symmetric metric on S 3 deformed along the fib ers of the Hopf fibration and then transferred to L ( p ; q ). More precisely , w e consider a SU 1 -left-in v arian t and U 1 -righ t-in v arian t Riemannian metric on S 3 , in other words, a left-inv ariant Riemannian metric g on the Lie group SU 2 that in the tangent space of the identit y reads as g ( u 1 , u 2 , u 3 ) = I 1 u 2 1 + I 1 u 2 2 + I 3 u 2 3 , where ( u 1 , u 2 , u 3 ) ∈ su 2 = T id SU 2 , and u 1 , u 2 , u 3 are co ordinates in the Lie algebra su 2 corresp onding to the basis e 1 = 1 2  0 1 − 1 0  , e 2 = 1 2  0 i i 0  , e 3 = 1 2  i 0 0 − i  . Remark 2. Notice that if I 1 = I 3 , then g = 4 I 1 s , where s is the standard metric on the unit sphere S 3 ⊂ R 3 (i.e., restricted from the Euclidian structure of R 3 ). Prop osition 2. The gr oup Z p acts on S 3 ≃ SU 2 by isometries. Pr o of. Define b y f : S 3 → S 3 the action of the generator of Z p , i.e., f : ( z , w ) 7→ ( εz , ε q w ), where ε = e i 2 π p . It is easy to chec k that d f ◦ dL ( z ,w ) ( iξ , ζ ) = dL f ( z ,w ) ( iξ , ε q − 1 ζ ) , where L ( z ,w ) denotes the left-shift by an elemen t  z w − ¯ w ¯ z  ∈ SU 2 and ( iξ , ζ ) is a tangent vector to SU 2 at the iden tity point, i.e.,  iξ ζ − ¯ ζ − iξ  ∈ su 2 . Since our metric is axisymmetric, i.e., U 1 -in v arian t, then ( iξ , ζ ) 7→ ( iξ , ε q − 1 ζ ) is lo cal isometry . 3 Prop osition 2 implies that w e can transfer axisymmetric Riemannian metric from SU 2 to the lens space L ( p ; q ). W e call the resulting metric the Ber ger metric on L ( p ; q ) by analogue with the Berger sphere. Let as in tro duce a parameter that measures the oblateness of the metric: η = I 1 I 3 − 1 > 0 . When η → − 1, or, equiv alen tly , I 3 → + ∞ , we obtain sub-Riemannian structure on L ( p ; q ). The corresp onding sub-Riemannian distribution is the pro jection of the left-in v arian t distribution on the Lie group SU 2 defined by the subspace span { e 1 , e 2 } ⊂ su 2 . This sub-Riemannian structure w as studied b y U. Boscain and F. Rossi [ 8 , Sec. 4]. 3 Some necessary definitions and kno wn results An y geo desic in the lens space is a pro jection of a geo desic in the group SU 2 . W e use the Hamiltonian approac h to describ e geo desics in SU 2 , see [ 1 , Sec. 2] for details. Consider the cotangen t bundle π : T ∗ SU 2 → SU 2 . An y geo desic is a pro jection of a tra jectory of the Hamiltonian v ector field  H in T ∗ SU 2 corresp onding to the Hamiltonian H = 1 2  h 2 1 I 1 + h 2 2 I 1 + h 2 3 I 3  , where h i ( λ ) = ⟨ dL π ( λ ) e i , · ⟩ , λ ∈ T ∗ SU 2 are linear on the fib ers of the cotangent bundle Hamiltonians. Any arclength parameterized geo desic starting from the p oint id ∈ SU 2 is determined by its initial co v ector h ∈ C ⊂ T ∗ id SU 2 ≃ su ∗ 2 , where C = { h ∈ su ∗ 2 | H ( h ) = 1 2 } is a lev el surface of the Hamiltonian H . Definition 3. (1) The exp onential map is the map Exp : C × R + → L ( p ; q ) , Exp( h, t ) = Π ◦ π ◦ e t  H h, ( h, t ) ∈ C × R + , where e t  H is the flo w of the Hamiltonian v ector field  H and Π : SU 2 → L ( p ; q ) is the factorization b y the Z p -action. (2) A pair of diffeomorphisms s : C × R + and S : L ( p ; q ) → L ( p ; q ), where s keeps the time, is called a symmetry of the exp onential map if Exp ◦ s = S ◦ Exp. (3) A p oin t m ∈ L ( p ; q ) is called a Maxwel l p oint if t w o differen t geo desics of the same length starting from the p oin t o = Π(id) meet one another at the p oin t m . (4) A critical v alue of the exp onential map is called a c onjugate p oint . (5) If a geo desic starting from the p oin t o is optimal up to the p oin t c on it and is not optimal after this p oin t, then the p oin t c is called a cut p oint along this geo desic. The set of the cut p oin ts along all geo desics starting from the p oin t o is called the cut lo cus Cut o . Usually Maxwell p oints app ear due to symmetries of the exp onential map. If Sym is some group of symmetries, then b y t Sym max w e denote the first time when a Maxwell p oin t corresp onding to symmetry app ears and call it the first Maxwel l time for the symmetry gr oup Sym. Analogically one can define the first c onjugate time and the cut time . The first Maxwell time, the first conjugate time and the cut time are the functions of initial co vector of a geo desic: t Sym max : C → R + ∪ { + ∞} , t conj : C → R + ∪ { + ∞} , t cut : C → R + ∪ { + ∞} . W e will find the set of first Maxwell p oin ts corresp onding to symmetries M Sym o and pro ve that the exp onen tial map is a diffeomorphism of the following domains: Exp : { ( h, t ) ∈ C × R + | 0 < t < t Sym max ( h ) } → L ( p ; q ) \  cl M Sym o ∪ { o }  , where cl M Sym o is the closure of M Sym o . This will imply that Cut o = cl M Sym o . T o pro v e this diffeomor- phism we will need the inequality t Sym max ( h ) ⩽ t conj ( h ) for any h ∈ C . Also we need geo desic equations and the equations for the conjugate time obtained in [ 1 , 12 ]. 4 In tro duce the follo wing notation. If h = ( h 1 , h 2 , h 3 ) ∈ su ∗ 2 are co ordinates in the basis dual to the basis e 1 , e 2 , e 3 ∈ su 2 , then | h | = q h 2 1 + h 2 2 + h 2 3 , ¯ h i = h i | h | , i = 1 , 2 , 3 , τ = 2 I 1 t | h | . It is w ell known that geo desics in SU 2 are pro ducts of tw o one-parametric subgroups in our case, but we need an explicit formulas in co ordinates. Prop osition 3 ([ 1 ], form ula (4)) . A ge o desic starting fr om the p oint id ∈ SU 2 with initial c ove ctor h ∈ C ⊂ su ∗ 2 has the fol lowing p ar ametrization. If z = q 0 + iq 3 and w = q 1 + iq 2 , then q 0 ( τ ) = cos τ cos ( τ η ¯ h 3 ) − ¯ h 3 sin τ sin ( τ η ¯ h 3 ) ,  q 1 ( τ ) q 2 ( τ )  = sin τ R − τ η ¯ h 3  ¯ h 1 ( τ ) ¯ h 2 ( τ )  , q 3 ( τ ) = cos τ sin ( τ η ¯ h 3 ) + ¯ h 3 sin τ cos ( τ η ¯ h 3 ) , wher e R α is the matrix of r otation by the angle α . Prop osition 4 ([ 12 ]) . The c onjugate time is e qual to t conj ( h ) = 2 I 1 τ conj ( ¯ h 3 ) | h | , wher e τ conj ( ¯ h 3 ) is the mini- mum value of π and the smal lest p ositive r o ot of the e quation − τ η (1 − ¯ h 2 3 ) cos τ − (1 + η ¯ h 2 3 ) sin τ = 0 . (1) (1) If − 1 < η ⩽ 0 , then τ conj ( ¯ h 3 ) = π for any ¯ h 3 ∈ [ − 1 , 1] . (2) If η > 0 , then τ conj ( ¯ h 3 ) is the smal lest p ositive r o ot of the e quation ( 1 ) and the ine quality π 2 < τ conj ( ¯ h 3 ) ⩽ π is satisfie d. Ther e is the e quality only for ¯ h 3 = ± 1 . 4 Maxw ell strata The exp onential map for an axisymmetric Riemannian problem on the lens space L ( p ; q ) has the same symmetry group as in the cases of SU 2 and SO 3 . The group of symmetries is Sym ≃ O 2 × Z 2 , we refer for details to [ 1 , Sec. 4]. The generators of this group act in the pre-image and in the image of the exp onen tial map in the follo wing wa y: (s1) s is a rotation around h 3 -axis and S is a rotation around q 3 -axis in ( q 1 , q 2 , q 3 )-space; (s2) s is a reflection with resp ect to a plane containing h 3 -axis and S is a reflection with resp ect to a plane containing q 3 -axis; (s3) s is the reflection with respect to the ( h 1 , h 2 )-plane and S is the reflection with respect to the ( q 1 , q 2 )-plane. It is already kno wn that the first Maxw ell time t Sym max for p > 1 dep ends on the Maxw ell time corresp onding to the rotations (s1) and comp ositions of the rotations (s1) with the reflection (s3), see [ 1 , Prop. 4]. It is easy to see from the geo desic equations (Prop osition 3 ) that the Maxwell time for rotations corresp onds to τ = π , while the comp ositions of the rotations by angles 2 π kq p with the reflection (s3) giv e Maxw ell p oin ts that are determined b y top ology of the lens space L ( p ; q ). These Maxw ell points form the surface ∂ L with iden tified p oints as describ ed in Proposition 1 . The corresponding τ for this Maxw ell time is the first p ositive ro ot with resp ect to the v ariable τ of the equation of the surface ∂ L where the parametric geo desics equations are substituted. Thus, the comparison of this first p ositive ro ot with π pla ys a k ey role in the analysis of Maxwell p oin ts. Remem b er that the surface ∂ L ⊂ S 3 for p > 1 has the equation q 2 1 + q 2 2 + q 2 3 sin 2 π p − 1 = − q 2 0 − q 2 3 + q 2 3 sin 2 π p = 1 sin 2 π p  q 2 3 cos 2 π p − q 2 0 sin 2 π p  = = 1 sin 2 π p  q 3 cos π p − q 0 sin π p   q 3 cos π p + q 0 sin π p  = 0 . 5 Figure 2: The functions τ − ℓ , τ + ℓ : [0 , 1] → R + . Figure 3: The function τ − ℓ and π in the case η < − p − 1 p . Let us introduce the function ℓ ( q ) = ℓ − ( q ) ℓ + ( q ), where ℓ − ( q ) = q 3 cos π p − q 0 sin π p = cos τ sin  τ η ¯ h 3 − π p  + ¯ h 3 sin τ cos  τ η ¯ h 3 − π p  , ℓ + ( q ) = q 3 cos π p + q 0 sin π p = cos τ sin  τ η ¯ h 3 + π p  + ¯ h 3 sin τ cos  τ η ¯ h 3 + π p  . (2) Denote b y τ ℓ the first p ositive ro ot of ℓ ( q 0 ( ¯ h 3 , τ ) , q 1 ( ¯ h 3 , τ ) , q 2 ( ¯ h 3 , τ ) , q 3 ( ¯ h 3 , τ )) with resp ect to v ariable τ and fixed ¯ h 3 , Ob viously , τ ℓ = min ( τ − ℓ , τ + ℓ ), where τ − ℓ and τ + ℓ are the first p ositiv e ro ots of the equations (resp ectiv ely): ℓ − ( q 0 ( ¯ h 3 , τ ) , q 1 ( ¯ h 3 , τ ) , q 2 ( ¯ h 3 , τ ) , q 3 ( ¯ h 3 , τ )) = 0 , ℓ + ( q 0 ( ¯ h 3 , τ ) , q 1 ( ¯ h 3 , τ ) , q 2 ( ¯ h 3 , τ ) , q 3 ( ¯ h 3 , τ )) = 0 . W e regard τ − ℓ and τ + ℓ as functions of ¯ h 3 , so τ − ℓ , τ + ℓ : [ − 1 , 1] → R + . Remark 3. Note that if p = 1, then ℓ ( q ) = q 2 3 and τ ℓ ( ¯ h 3 ) = τ − ℓ ( ¯ h 3 ) = τ + ℓ ( ¯ h 3 ) is the first p ositiv e ro ot of the equation q 3 ( ¯ h 3 , τ ) = 0. This Maxw ell time corresp onding to the reflection with resp ect to the plane h 3 = 0, i.e., the symmetry (s3), plays the k ey role in the Maxwell time analysis in the case p = 1, see [ 1 , Prop. 8–10]. So, the notation τ ℓ is universal and do esn’t dep end on p , while the surface ℓ = 0 is the surface of fixed p oint for different symmetries that dep end on p . Lemma 1. The fol lowing expr ession for τ ℓ is satisfie d : τ ℓ ( ¯ h 3 ) =  τ − ℓ ( ¯ h 3 ) , if ¯ h 3 ⩾ 0 , τ + ℓ ( ¯ h 3 ) , if ¯ h 3 < 0 . Pr o of. Note that ℓ − ( − ¯ h 3 ) = − ℓ + ( ¯ h 3 ). This implies that τ − ℓ ( − ¯ h 3 ) = τ + ℓ ( ¯ h 3 ), see Fig. 2 . If p = 1 or p = 2, then τ − ℓ ( ¯ h 3 ) = τ + ℓ ( ¯ h 3 ). Assume that p > 2. W e hav e τ − ℓ (0) = τ + ℓ (0) = π 2 and τ − ℓ (1) = π p (1+ η ) < ( p − 1) π p (1+ η ) = τ + ℓ (0). If τ − ℓ ( ¯ h 3 ) = τ + ℓ ( ¯ h 3 ) for some ¯ h 3 , then q 0 ( ¯ h 3 ) = q 3 ( ¯ h 3 ) = 0, this implies that ¯ h 3 = 0. So, τ − ℓ ( ¯ h 3 ) ⩽ τ + ℓ ( ¯ h 3 ) for ¯ h 3 ⩾ 0 and τ − ℓ ( ¯ h 3 ) ⩾ τ + ℓ ( ¯ h 3 ) for ¯ h 3 < 0. Prop osition 5. (1) If η ⩾ − p − 1 p , then τ ℓ ( ¯ h 3 ) ⩽ π for any ¯ h 3 ∈ [ − 1 , 1] . (2) If η < − p − 1 p , then τ ℓ ( ¯ h 3 ) ⩾ π for p − 1 p | η | ⩽ | ¯ h 3 | ⩽ 1 and τ ℓ ( ¯ h 3 ) < π for | ¯ h 3 | < p − 1 p | η | , se e Fig. 3 . Pr o of. Note that since q 0 | τ =0 = 1 and q 3 | τ =0 = 0, then ℓ − | τ =0 = − 1 < 0 and ℓ + | τ =0 = 1 > 0. So, to prov e (1) and the second part of (2) it is sufficien t to find a function θ η : [ − 1 , 1] → R suc h that θ η ( ¯ h 3 ) ⩽ π and ℓ − | τ = θ η ( ¯ h 3 ) ⩾ 0 or ℓ + | τ = θ η ( ¯ h 3 ) ⩽ 0. Indeed, since the functions ℓ − and ℓ + of v ariable τ are contin uous, then the function ℓ has a ro ot on the segment (0 , θ η ( ¯ h 3 )] ⊂ (0 , π ]. Let us choose θ η ( ¯ h 3 ) = ( π , if | η ¯ h 3 | ⩽ p − 1 p , ( p − 1) π p | η ¯ h 3 | , if | η ¯ h 3 | > p − 1 p . 6 If | η ¯ h 3 | ⩽ p − 1 p , then ℓ − | τ = θ η ( ¯ h 3 ) = ℓ − | τ = π = − sin  π η ¯ h 3 − π p  , ℓ + | τ = θ η ( ¯ h 3 ) = ℓ + | τ = π = − sin  π η ¯ h 3 + π p  . If − p − 1 p ⩽ η ¯ h 3 ⩽ 1 p , then − π ⩽ π η ¯ h 3 − π p ⩽ 0 and ℓ − | τ = θ η ( ¯ h 3 ) = − sin  π η ¯ h 3 − π p  ⩾ 0. If − 1 p ⩽ η ¯ h 3 ⩽ p − 1 p , then 0 ⩽ π η ¯ h 3 + π p ⩽ π and ℓ + | τ = θ η ( ¯ h 3 ) = − sin  π η ¯ h 3 + π p  ⩽ 0. Consider now the case | η ¯ h 3 | > p − 1 p and η > 0. By form ula ( 2 ) w e ha v e ¯ h 3 < 0 ⇒ ℓ − | τ = θ η ( ¯ h 3 ) = − ¯ h 3 sin ( p − 1) π p | η ¯ h 3 | > 0 , ¯ h 3 > 0 ⇒ ℓ + | τ = θ η ( ¯ h 3 ) = − ¯ h 3 sin ( p − 1) π p | η ¯ h 3 | < 0 , This implies (1) and the second part of (2). It remains to consider the case | η ¯ h 3 | > p − 1 p and − 1 < η < − p − 1 p . Let us pro ve that τ ℓ ( ¯ h 3 ) > π for | ¯ h 3 | > p − 1 p | η | . It is not difficult to see that ℓ − | ¯ h 3 = ± 1 = sin  ± τ (1 + η ) − π p  ⇒ τ − ℓ (1) = π p (1+ η ) > π , τ − ℓ ( − 1) = ( p − 1) π p (1+ η ) > π , ℓ + | ¯ h 3 = ± 1 = sin  ± τ (1 + η ) + π p  ⇒ τ + ℓ (1) = ( p − 1) π p (1+ η ) > π , τ + ℓ ( − 1) = π p (1+ η ) > π . This means that τ ℓ ( ± 1) > π . Assume by contradiction that there exists ˆ h 3 suc h that p − 1 p | η | < | ˆ h 3 | < 1 and ℓ ± | ¯ h 3 = ˆ h 3 ,τ = π = − sin  π η ˆ h 3 ± π p  = 0 . Solving this equation with resp ect to ˆ h 3 , we get | ˆ h 3 | = | kp ± 1 | p | η | where k ∈ Z . But any p oint of this series lies outside the in terv al  p − 1 p | η | , 1  . So, we get a contradiction. 5 The Maxw ell time is less or equal to the conjugate time W e need the follo wing technical prop osition. Prop osition 6. The functions τ − ℓ and τ + ℓ ar e c ontinuous on the interval [ − 1 , 1] . Pr o of. W e may prov e this statement only for τ − ℓ thanks to Lemma 1 . It is sufficient to pro ve that for any ¯ h 3 ∈ [ − 1 , 1] there is no τ suc h that ℓ − | ¯ h 3 ,τ = 0 and ∂ ℓ − ∂ τ | ¯ h 3 ,τ = 0. Assume by contradiction that there is suc h τ . F rom ( 2 ) we obtain ℓ − | ¯ h 3 ,τ = cos τ sin  τ η ¯ h 3 − π p  + ¯ h 3 sin τ cos  τ η ¯ h 3 − π p  = 0 , ∂ ℓ − ∂ τ | ¯ h 3 ,τ = − (1 + η ¯ h 2 3 ) sin τ sin  τ η ¯ h 3 − π p  + ¯ h 3 (1 + η ) cos τ cos  τ η ¯ h 3 − π p  = 0 . (3) Consider the case ¯ h 3 = 0. It follo ws that cos τ = sin τ = 0 and we get a contradiction. So, w e can assume that ¯ h 3  = 0. Assume that cos τ cos τ cos  τ η ¯ h 3 − π p  = 0. If cos τ = 0, then since ¯ h 3  = 0 and sin τ  = 0, from the first equation w e get cos  τ η ¯ h 3 − π p  = 0. Whence, from the second equation since sin  τ η ¯ h 3 − π p   = 0 and 1 + η ¯ h 2 3 > 0 we get sin τ = 0. W e get a contradiction. The same arguments imply that cos  τ η ¯ h 3 − π p   = 0. So, we ma y divide b oth equations b y cos τ cos τ cos  τ η ¯ h 3 − π p  . W e obtain tan  τ η ¯ h 3 − π p  + ¯ h 3 tan τ = 0 , − (1 + η ¯ h 2 3 ) tan τ tan  τ η ¯ h 3 − π p  + ¯ h 3 (1 + η ) = 0 . 7 It follows that (1 + η ¯ h 2 3 ) ¯ h 3 tan 2 τ + ¯ h 3 (1 + η ) = 0 , ¯ h 3  = 0 ⇒ (1 + η ¯ h 2 3 ) tan 2 τ = − (1 + η ) , but 1 + η > 0 and 1 + η ¯ h 2 3 > 0. W e get a contradiction. Prop osition 7. The first Maxwel l time is less or e qual to the c onjugate time. Pr o of. F rom Prop osition 5 we know that t Sym max ( ¯ h 3 ) = 2 I 1 τ ℓ ( ¯ h 3 ) | h | , for η > − p − 1 p , t Sym max ( ¯ h 3 ) = ( 2 I 1 π | h | , if | ¯ h 3 | ⩾ p − 1 p | η | , 2 I 1 τ ℓ ( ¯ h 3 ) | h | , if | ¯ h 3 | < p − 1 p | η | , for − 1 < η < − p − 1 p . Hence, from Lemma 1 and Prop osition 4 it follo ws that it is sufficien t to pro v e that for η > 0 the inequalit y τ − ℓ ( ¯ h 3 ) ⩽ t conj ( ¯ h 3 ) is satisfied for ¯ h 3 ⩾ 0. W e consider tw o cases: p = 1 and p > 1. F or the case p = 1 we refer to [ 1 , Prop. 10]. Assume now that p > 1. Thanks to Prop osition 5 it is sufficient to sho w that τ − ℓ ( ¯ h 3 ) ⩽ π 2 for ¯ h 3 ⩾ 0. First, note that the first p ositive ro ot of the equation ℓ − | ¯ h 3 =1 = sin  τ (1 + η ) − π p  = 0 equals τ − ℓ (1) = π p (1+ η ) < π 2 ⩽ π 2 . Assume that there exists ¯ h 3 ∈ [0 , 1] suc h that τ − ℓ ( ¯ h 3 ) > π 2 . Since the function τ − ℓ is contin uous by Prop osition 6 , then there exists ˆ h 3 ∈ ( ¯ h 3 , 1) such that τ − ℓ ( ˆ h 3 ) = π 2 . This mean that ℓ − | ¯ h 3 = ˆ h 3 ,τ = π 2 = ˆ h 3 cos π η ˆ h 3 2 − π p ! = 0 ⇒ ˆ h 3 = (2 k + 1) p + 2 pη , k ∈ Z . Compute the deriv ativ e of the function ℓ − at these p oin ts using form ula ( 3 ) ∂ ℓ − ∂ τ | ¯ h 3 = ˆ h 3 ,τ = π 2 = − (1 + η ˆ h 2 3 ) sin π η ˆ h 3 2 − π p ! . Since cos  π η ˆ h 3 2 − π p  = 0 and 1 + η ¯ h 3 > 0, then the signs of deriv ativ es at these p oint alternate. Second, consider the arc of the graph of the function τ − ℓ b et ween some neigh b our p oints. Since the function ℓ − is smo oth there exists a p oin t ( ¯ h 3 , τ − ℓ ( ¯ h 3 )) on this arc such that ∂ ℓ − ∂ τ = 0 at this p oin t. Moreo v er, ℓ − = 0 at this point as w ell. But this is not p ossible as follows from the pro of of Prop osition 6 , see ( 3 ). W e get a contradiction. 6 The cut lo cus and the cut time Theorem 1. (1) The cut time with r esp e ct to the initial p oint o = Π(id) for the Ber ger lens sp ac e L ( p ; q ) is e qual to the first Maxwel l time, i.e., t cut ( ¯ h 3 ) = 2 I 1 τ ℓ ( ¯ h 3 ) | h | , for η > − p − 1 p , t cut ( ¯ h 3 ) = ( 2 I 1 π | h | , if | ¯ h 3 | ⩾ p − 1 p | η | , 2 I 1 τ ℓ ( ¯ h 3 ) | h | , if | ¯ h 3 | < p − 1 p | η | , for − 1 < η < − p − 1 p . Mor e over, τ ℓ ( ¯ h 3 ) =  τ − ℓ ( ¯ h 3 ) , if ¯ h 3 ⩾ 0 , τ + ℓ ( ¯ h 3 ) , if ¯ h 3 < 0 . (2) The cut lo cus Cut o with r esp e ct to the initial p oint o = Π(id) of the Ber ger lens sp ac e L ( p ; q ) , p > 1 is e qual to 8 (a) ∂ L/ ∼ for η ⩾ − p − 1 2 ( se e Pr op osition 1 for the definition of the e quivalenc e r elation ∼ ); (b) the we dge sum of ∂ L/ ∼ and the interval h − sin π p , − sin ( − π η ) i ∪ h sin ( − π η ) , sin π p i on the q 3 -axis, se e Fig. 4 . Figure 4: The cut locus for axisymmetric Rie- mannian metric on the lens space L ( p ; q ) for p > 1 and η < − p − 1 p has tw o strata ∂ L/ ∼ and an interv al. Figure 5: The cut locus for axisymmetric sub- Riemannian metric on the lens space L ( p ; q ) for p > 1 has tw o strata ∂ L/ ∼ and a punctured circle. This corresp onds to η → − 1. Remark 4. Note that the additional stratum of the cut lo cus in item (2b) of Theorem 1 is an in terv al since in the mo del of the lens space L ( p ; q ) describ ed in Proposition 1 w e should iden tify the p oints ± sin π p on the q 3 -axis (the North and the South p oles of the ellipsoid of revolution). Remark 5. If q = 1, then the lens space L ( p ; 1) is homogeneous and the structure of the cut time and the cut lo cus do esn’t dep end on the initial p oint. F or q > 1 it is significan t that w e consider an initial p oint o = Π(id). F or the top ological structure of the t wo dimensional comp onent of the cut lo cus w e refer to the w ork of S. Anisov [ 9 ] were the cut lo cus for symmetric ( I 1 = I 3 ) Riemannian metric is studied. Note that the cut lo cus for symmetric metric coincides with tw o dimensional comp onent ∂ L/ ∼ of the cut lo cus in our case. In pap er [ 9 ], the cut lo cus for arbitrary initial p oint for symmetric metric is also describ ed. Remark 6. Theorem 1 agrees with the results obtained in [ 1 , Th. 5, Th. 3] for L (1; 1) = SU 2 and L (2; 1) = SO 3 . Remark 7. If p = 1, then the surface ℓ ( q ) = q 2 3 = 0 is a t w o dimensional sphere S 2 = { q 2 0 + q 2 1 + q 2 2 = 1 } , see Remark 3 . In this case for η > 0 not the whole sphere is a Maxw ell set and consequently the cut lo cus, but only a tw o dimensional disk on this sphere. The b oundary of this disk consists of conjugate p oin ts. This is the result of T. Sak ai [ 14 ] for the cut lo cus of the Berger sphere in the case η > 0. The reason of this phenomenon is that for a p oint on this sphere outside the disk there is only one geo desic coming to this p oin t in time less than the cut time, since this Maxwell set (the disk) corresp onds to the reflection with resp ect to the plane h 3 = 0 and this symmetry has stationary p oints on this plane h 3 = 0 in con trast with Z p -symmetries providing the Maxw ell strata for p > 1. Pr o of of The or em 1 . First, the expression for τ ℓ follo ws from Lemma 1 . Second, let us find the first Maxw ell p oints corresp onding to rotations, i.e., the first Maxwell time with τ = π for η < − p − 1 p and | ¯ h 3 | ⩾ p − 1 p | η | . Substituting these v alues to geodesic equations of Prop osition 3 w e get the segment (2b). While the first Maxwell p oint corresp onding to the first Maxwell time defined b y τ ℓ giv e the t w o dimensional comp onen t ∂ L/ ∼ . So, we hav e describ ed the set of the first Maxwell p oin ts M Sym o . No w it is sufficient to pro ve that the exp onential map is a diffeomorphism Exp : U = { ( h, t ) ∈ C × R + | 0 < t < t Sym max ( h ) } → L ( p ; q ) \  cl M Sym o ∪ { o }  . W e use the Hadamard global diffeomorphism theorem [ 13 ]. A smo oth non-degenerate prop er map of t w o connected and simply connected manifolds of same dimensions is a diffeomorphism. Indeed, 9 these tw o domains are three dimensional connected and simply connected manifolds. Moreo v er, the exp onen tial map is smo oth and non-degenerate, since the conjugate time is greater than or equal to the first Maxw ell time due to Prop osition 7 . It remains to sho w that our map is prop er, i.e., for any compact K ⊂ L ( p ; q ) \  cl M Sym o ∪ { o }  the set Exp − 1 K is compact as w ell. It is sufficient to prov e that Exp − 1 K is closed. Assume b y contradiction that Exp − 1 K is not closed. Then there exists a sequence ( h n , t n ) that con v erges to some ( h, t ) ∈ cl U \ Exp − 1 K . Since Exp is a contin uous map, then Exp ( h n , t n ) conv erges to Exp ( h, t ) ∈ K , since K is compact. If ( h, t ) ∈ U , then ( h, t ) ∈ Exp − 1 K , we get a contradiction. Consider no w the case ( h, t ) ∈ ∂ U . This means that t = 0 or t equals the first Maxwell time t Sym max ( h ). Hence, Exp ( p, t ) ∈ cl M Sym o ∪ { o } , but K is compact. Theorem 2. The cut time and the cut lo cus for the Ber ger lens sp ac e L ( p ; q ) ( with r esp e ct to the initial p oint o = Π(id)) c onver ge to the cut time and the cut lo cus of the axisymmetric sub-R iemannian structur e on L ( p ; q ) when η → − 1 ( or, e quivalent, I 3 → + ∞ ) , se e Fig. 5 . Pr o of. It is a direct computation. W e refer for the cut time and the cut lo cus of the sub-Riemannian structure to pap er [ 8 , Th. 4]. Note that the segmen t (2b) in Theorem 1 is closing up to a punctured circle while η → − 1. 7 Lo w er b ound for diameter In this Section, we find a low er b ound for diameter of an axisymmetric Riemannian metric on a lens space. More precisely , w e find the maximum of distances from the p oint o to other p oints. Ob viously , this maximum equals to the maximum of the cut time. Moreov er, for lens spaces L ( p ; 1) this is the exact v alue of the diameter, since these spaces are homogeneous. So, w e need to study the cut time as a function of v ariable ¯ h 3 . W e need several tec hnical lemmas. Lemma 2. The cut time t cut : [0 , 1] → R + is an even function of variable ¯ h 3 . Pr o of. It follows from formulas ( 2 ) that the function ℓ = ℓ − ℓ + is an even function of the v ariable ¯ h 3 . Th us, its first p ositive ro ot τ ℓ is even to o (see also Lemma 1 ). F or | h | as a function of the v ariable ¯ h 3 w e hav e h 2 1 + h 2 2 I 1 + h 2 3 I 3 = 1 ⇒ | h | = √ I 1 p 1 + η ¯ h 2 3 . (4) Hence, | h | is ev en and using the expression of the cut time from Theorem 1 (1) we obtain that the cut time is an ev en function as well. No w we assume that ¯ h 3 ∈ [0 , 1] without loss of generalit y . Lemma 3. (1) If − 1 < η < − p − 1 p , then the function t cut has one critic al p oint on the interval  0 , p − 1 p | η |  . Mor e over, it is a p oint of minimum. (2) If − p − 1 p ⩽ η < 0 , then the function t cut has no critic al p oints on the interval (0 , 1) . (3a) If p = 1 and η = 0 , then t cut ( ¯ h 3 ) = 2 π √ I 1 = const . (3b) If p = 2 and η = 0 , then t cut ( ¯ h 3 ) = π √ I 1 = const . (3c) If p > 2 and η = 0 , then t cut ( ¯ h 3 ) = ( 2 I 1 arctan  1 ¯ h 3 tan π p  , if ¯ h 3  = 0 , I 1 π , if ¯ h 3 = 0 . (4a) If p = 1 and 0 < η < 1 , then ther e ar e no critic al p oints of the function t cut on the interval (0 , 1) . (4b) If p > 1 and 0 < η < 2 p , then ther e ar e no critic al p oints of the function t cut on the interval (0 , 1) . 10 (5a) If p = 1 and 1 < η , then ther e is only one critic al p oint of the function t cut on the interval (0 , 1) . This p oint is ¯ h 3 = 1 η , it is a p oint of maximum. (5b) If p > 1 and 2 p < η , then ther e is only one critic al p oint of the function t cut on the interval (0 , 1) . Mor e over, it is a p oint of minimum. Pr o of. Let us compute the deriv ativ e of the function t cut , we get dt cut d ¯ h 3 = 2 p I 1 dτ − ℓ d ¯ h 3 q 1 + η ¯ h 2 3 + 2 p I 1 τ − ℓ ( ¯ h 3 ) η ¯ h 3 p 1 + η ¯ h 2 3 , dτ − ℓ d ¯ h 3 = − ∂ ℓ − ∂ ¯ h 3 . ∂ ℓ − ∂ τ . Using formula ( 2 ) w e obtain ∂ ℓ − ∂ ¯ h 3 = τ η cos τ cos  τ η ¯ h 3 − π p  + sin τ cos  τ η ¯ h 3 − π p  − τ η ¯ h 3 sin τ sin  τ η ¯ h 3 − π p  . Then from the second formula in ( 3 ) up to the p ositiv e multiplier 2 √ I 1 / p 1 + η ¯ h 2 3 w e get dt cut d ¯ h 3 ∼ cos  τ − ℓ η ¯ h 3 − π p   − τ − ℓ η (1 − ¯ h 2 3 ) cos τ − ℓ − (1 + η ¯ h 2 3 ) sin τ − ℓ  − (1 + η ¯ h 2 3 ) sin τ − ℓ sin  τ − ℓ η ¯ h 3 − π p  + ¯ h 3 (1 + η ) cos τ − ℓ sin  τ − ℓ η ¯ h 3 − π p  , Notice that the expression in the square brack ets in the nominator equals to the expression in the equation for the conjugate time ( 1 ). Since the first Maxwell time is less than the first conjugate time on in terv als under consideration (see Prop osition 7 ), then the sign of the expression in the square brack ets coincides with the sign of this expression when τ = 0 which is negative. Hence, the critical p oints of the function τ − ℓ are determined by the equations cos  τ − ℓ η ¯ h 3 − π p  = 0 and ℓ − ( τ − ℓ ) = 0 ⇒ cos τ − ℓ = 0 . Whence, τ − ℓ = π 2 and ¯ h 3 = (2 k +1) p +2 pη , where k ∈ Z . (1) The only p oin t of this series lying on the interv al  0 , p − 1 p | η |  is the p oin t ¯ h 3 = p − 2 p | η | when k = − 1. Since lim ¯ h 3 → 0 dt cut d ¯ h 3 < 0 this is a p oint of minim um, see Fig. 6 (a). (2) If η ⩾ − p − 1 p , then there are no p oin ts of the series kp +2 pη lying in the in terv al (0 , 1). (3) If η = 0, then | h | = 1 and this result can b e obtained by the direct computation of the first p ositiv e ro ot of the function ℓ − . (4) Assume that η > 0. It is easy to see that an y p oin t of the series kp +2 pη is outside the interv al (0 , 1) if and only if 0 < η < 1 for p = 1 and 0 < η < 2 p for p > 1. (5) Let us prov e that the function τ − ℓ decreases on the interv al (0 , 1) for η > 0. Compute the deriv ative dτ − ℓ d ¯ h 3 = − ∂ ℓ − ∂ ¯ h 3 . ∂ ℓ − ∂ τ − ℓ = = − τ − ℓ η cos τ − ℓ cos  τ − ℓ η ¯ h 3 − π p  + sin τ − ℓ cos  τ − ℓ η ¯ h 3 − π p  − τ − ℓ η ¯ h 3 sin τ − ℓ sin  τ − ℓ η ¯ h 3 − π p  − (1 + η ¯ h 2 3 ) sin τ − ℓ sin  τ − ℓ η ¯ h 3 − π p  + ¯ h 3 (1 + η ) cos τ − ℓ cos  τ − ℓ η ¯ h 3 − π p  . Since ℓ − ( τ − ℓ ) = 0 w e get that if cos τ − ℓ = 0, then cos  τ − ℓ η ¯ h 3 − π p  = 0 for ¯ h 3  = 0. Moreo v er, if cos  τ − ℓ η ¯ h 3 − π p  = 0, then cos τ − ℓ = 0. If cos τ − ℓ cos  τ − ℓ η ¯ h 3 − π p  = 0, then it is easy to see that dτ − ℓ d ¯ h 3 < 0. 11 So, we ma y assume that cos τ − ℓ cos  τ − ℓ η ¯ h 3 − π p   = 0 and divide the nominator and the denominator b y this expression. W e obtain dτ − ℓ d ¯ h 3 = − τ − ℓ η + tan τ − ℓ − τ − ℓ η ¯ h 3 tan τ − ℓ tan  τ − ℓ η ¯ h 3 − π p  − (1 + η ¯ h 2 3 ) tan τ − ℓ tan  τ − ℓ η ¯ h 3 − π p  + ¯ h 3 (1 + η ) . But ℓ − ( τ − ℓ ) = 0 implies that tan  τ − ℓ η ¯ h 3 − π p  = − ¯ h 3 tan τ − ℓ . Hence, we get dτ − ℓ d ¯ h 3 = − τ − ℓ η + tan τ − ℓ + τ − ℓ η ¯ h 2 3 tan 2 τ − ℓ ¯ h 3 (1 + η ¯ h 2 3 ) tan 2 τ − ℓ + ¯ h 3 (1 + η ) < 0 for ¯ h 3 > 0 . It follows that the function τ − ℓ decreases on the in terv al (0 , 1). So, there exists almost one p oin t ¯ h 3 ∈ (0 , 1) suc h that τ ℓ = π 2 and dt cut d ¯ h 3 ( ¯ h 3 ) = 0. If p = 1 this p oint exists for η ⩾ 1 and is equal to ¯ h 3 = 1 η with k = − 1. Since lim ¯ h 3 → 0 dt cut d ¯ h 3 > 0 for η > 0, this is a p oin t of maxim um, see Fig. 6 (b). If p > 1, then this p oint is a p oin t of minim um, since lim ¯ h 3 → 0 dt cut d ¯ h 3 < 0. Lemma 4. Assume that η ⩾ − p − 1 p . Then t cut (0) ⩾ t cut (1) iff η ⩾ 4 p 2 − 1 . This ine quality is satisfie d automatic al ly if p ⩾ 4 . Pr o of. First, | h | = √ I 1 for ¯ h 3 = 0 and | h | = √ I 1 √ 1+ η for ¯ h 3 = 1 by form ula ( 4 ). Second, τ − ℓ (0) = π 2 and τ − ℓ (1) = π p (1+ η ) due to formula ( 2 ). Third, t cut (0) = 2 I 1 τ − ℓ (0) | h | = π p I 1 , t cut (1) = 2 I 1 τ − ℓ (1) | h | = 2 p I 1 π p √ 1 + η . (5) Finally , t cut (0) ⩾ t cut (1) ⇔ p p 1 + η ⩾ 2 ⇔ η ⩾ 4 p 2 − 1 . This inequality is satisfied automatically if 4 p 2 − 1 ⩽ − p − 1 p , this condition is equiv alen t to p ⩾ 4. Theorem 3. (1) Ther e ar e the fol lowing lower b ounds for diameter of the Ber ger lens sp ac e L ( p ; q ) . (a) If − 1 < η < − p − 1 p , then diam L ( p ; q ) ⩾ ( 2 π √ I 1 q 1 + ( p − 1) 2 p 2 η , if p ∈ h 4 − √ − 12 η 3 η +4 , 4+ √ − 12 η 3 η +4 i , π √ I 1 , else . (b) If − p − 1 p ⩽ η < 0 , then diam L ( p ; q ) ⩾ ( 2 π √ I 3 p , if p < 2 √ 1+ η , π √ I 1 , else . (c) If η = 0 , then diam L ( p ; q ) ⩾  2 π √ I 1 , if p = 1 , π √ I 1 , if p ⩾ 2 . (d) If 0 < η and p > 1 , then diam L ( p ; q ) ⩾ ( 2 π √ I 3 p , if p < 2 √ 1+ η , π √ I 1 , else . 12 (a) η < − p − 1 p (b) p = 1 , η > 1 Figure 6: The cases where the cut time t cut : [0 , 1] → R + has maximum that differs from t cut (0) and t cut ( ± 1). (e) If 0 < η and p = 1 , then diam L ( p ; q ) ⩾  2 π √ I 3 , if 0 < η ⩽ 1 , π I 1 √ I 1 − I 3 , if 1 < η . (2) F or the Ber ger lens sp ac e L ( p ; 1) these b ounds ar e exact values of diameters. Remark 8. In the case of symmetric metric (i.e., η = 0), usually the standard metric s on the unit sphere S 3 ⊂ R 3 and the corresp onding metric on the lens space L ( p ; q ) are considered. Since our metric differs b y a multiplier (see Remark 2 ), then diameter of our metric is 2 √ I 1 times greater than diameter of the standard metric. Thus, our result of Theorem 3 (1c) agrees with [ 9 , Lemma 2.1]. Remark 9. These v alues of diameter for p = 1 and p = 2 coincides with the previous results, see [ 2 , Th. 1] and [ 1 , Th. 4], resp ectively . Remark 10. These diameter b ounds are contin uous as a functions of the v ariable η . Pr o of of The or em 3 . (1a) In this case the cut time as a function of v ariable ¯ h 3 is piecewise smooth, see Theorem 1 (1). Moreo ver, on the segment h p − 1 p | η | , 1 i this function decreases b y formula ( 4 ). There is one p oint of minimum on the interv al  0 , p − 1 p | η |  due to Lemma 3 (1). Hence, w e need to compare to v alu es: 2 I 1 τ − ℓ (0) | h | for ¯ h 3 = 0 and 2 I 1 π | h | for ¯ h 3 = p − 1 p | η | . The first one equals π √ I 1 b y ( 5 ) and the second one is 2 √ I 1 π q 1 + ( p − 1) 2 p 2 η . W e obtain π p I 1 ⩽ 2 p I 1 π s 1 + ( p − 1) 2 p 2 η ⇔ 1 ⩽ 4 + 4( p − 1) 2 p 2 η ⇔ 0 ⩽ (3 η + 4) p 2 − 8 p + 4 . This inequality is satisfied iff p ∈ h 4 − √ − 12 η 3 η +4 , 4+ √ − 12 η 3 η +4 i . (1b) Since b y Lemma 3 (2) there are no critical p oints of the function t cut , w e need to compare its v alu es at the ends of the in terv al [0 , 1], see Lemma 4 and formula ( 5 ). W e hav e π p I 1 < 2 π √ I 1 p √ 1 + η = 2 π √ I 3 p ⇔ p < 2 √ 1 + η . (1c) Immediately follows from Lemma 3 (3). (1d) Immediately follo ws from Lemma 3 (4b,5b) and the comparison of the v alues of the function t cut at the ends of the in terv al [0 , 1], see Lemma 4 . (1e) F ollo ws from Lemma 3 (4a,5a) and τ − ℓ ( 1 η ) = π 2 . (2) This is due to the fact that the lens space L ( p ; 1) is homogeneous. 13 References [1] P o dobryaev, A. V., Sachk o v, Y u. L.: Cut lo cus of a left in v arian t Riemannian metric on S O (3) in the axisymmetric case. Journal of Geometry and Physics. 110, 436–453 (2016) [2] P o dobryaev, A. V.: Diameter of the Berger Sphere. Mathematical Notes. 103, 5, 846–851 (2018) [3] P o dobryaev, A. V., Sac hk o v, Y u. 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