Algebraically closed real geodesics on n-dimensional ellipsoids are dense in the parameter space and related to hyperelliptic tangential coverings

Algebraically closed real geo desics on n-dimensional ellipsoids are dense in the para meter space a nd related to h yp erelliptic tangen ti al co verings ∗ Simonetta Ab enda Dipartimen to di Matematica e CIRAM Univ ersit` a degli Studi di Bologna, Italy aben da@ciram.unib o.it Octob er 28, 2018 Abstract The c lo sedness condition for real geo desics o n n –dimensional ellipsoids is in general transcendental in the par ameters (semiax e s of the ellipsoid and con- stants of motio n). W e show that it is alg ebraic in the para meters if a nd only if bo th the real and the ima ginary geo desics are closed and we character iz e such do uble–p erio dicity condition via real hyperelliptic tangen tial cov ering s. W e prove the density of algebr aically closed geo desics on n –dimensional ellip- soids with resp ect to the natural top ology in the 2 n –dimensio nal real parameter space. In particular, the approximating s e quence of algebraic clo sed geo desics on the approximated ellipsoids ma y b e c ho sen so to sha re the same v alues of the length and of the real p e rio d vector a s the limiting closed g e o desic o n the limiting ellipsoid. Finally , for rea l doubly–p erio dic geo desics on triax ial ellipsoids, we show how to ev aluate algebr aically the p er io d mapping and we present so me explicit examples of families of alg ebraically closed geo desics. 1 In tro d uction In tegrabilit y of the ge o desic motion on a tr iaxial ellipsoid Q was p r o ve n in 1838 by Jacobi [19] who r educed the system to hyp erelliptic quadr atur es; moreo ve r W eier- strass [40 ] in tegrated the sy s tem in term s of th eta–functions on a gen us 2 hyp er- elliptic curv e. The geo desic fl o w has man y interesti n g geometric prop erties: in particular, eac h geo desic on Q oscilla tes b et wee n the t wo lines of in tersection of Q with a confo cal hyp erb oloid Q c (caustic) and b y a theorem by C hasles [8] all the tangen t lines to the geo desics are also tangen t to Q c . The generic geo desic is quasi– p erio d ic and , in case a geo desic on Q is closed, then all the geo d esics on Q tangen t to the s ame confo cal hyp erb oloid are also closed and hav e the same length. ∗ AMS Su b ject Classification 37J35, 70H12,70H06 1 The theorem of Chasles generaliz es to n –dimensional quadrics Q and the set of common tangen t lin es to n confo cal qu ad r ics pla ys an im p ortant role in the study of th e ge o desics on an y of such quadr ics and in the reformulatio n of integ r ab ility of the system in the mo dern language of algebraically in tegrable systems (see Moser [27, 28], Kn ¨ orrer [21 , 22] and Audin [3]). In particular, in [22] Kn ¨ orrer settled the so–calle d Moser–T rub o witz isomorphism b et ween the geod esics on quadrics and the stationary solutions to the Kortew eg de V ries equation (Kd V). One of the consequences of Chasles theorem is that, wh en a geo desic on Q is closed, all the geo d esics sharin g th e s ame v alues of the constan ts of motion are closed and of the same length (see for instance [20]). The condition for a geo desic on an n –dimens ional qu adric Q to b e closed is then expressed as a certain linear com bination of inte grals of holomorph ic differentia ls on a h yp erelliptic curve. Su c h condition is tr an s cenden tal in the parameters of the pr oblem (semiaxes of the quadr ic Q and p arameters of the caustics) and, b y the Moser–T rub o witz isomorphism, it is equiv alen t to imp ose that the s tationary solutions of the KdV are real p er io dic in x . Characterization of the set for whic h the closedness prop ert y of real geo desics on n –dimensional ellipsoids is algebraic in the parameters A natural question is then: is it p ossible to settle extra conditions so that th e closedness prop erty (2.7) of the geo desic b e algebraic in the p arameters (semiaxes of the quadric Q and the constan ts of motions)? I n [13, 1], w e found a set of sufficien t conditions in th e complex setting: we introdu ced and c haracterized a family of algebraic closed geod esics asso ciated to h yp erelliptic tangen tial co ve r s. The results in the ab o ve pap ers indicate that the closedness pr op erty is algebraic (in the p arameters) if the p erio dicit y condition is essenti ally one–dimensional in the complex setting. In th e algebraic-geomet r ic setting, th is in turn means th at the closedness condition is algebraic (in the parameters) if it is equiv alen t to the in version of an elliptic in tegral. In the present pap er , we complete the c haracterization of algebraically closed geod esics: we restrict ourselv es to the real setting and we settle the necessary and sufficien t cond itions so th at the closedness pr op ert y b e algebraic in the real param- eters (semiaxes of th e ellipsoid and constant s of motio n ). In particular, we pr o ve that the closedness condition is algebraic in the p aram- eters if and only if b oth the real and the imaginary geo desics on the n –dimensional ellipsoid are closed. Th e double p erio dicity condition we introdu ce her e for the real geod esics on ellipsoids is mo delled after a similar condition b y Mc K ean and v an Mo erb eke[ 26 ] for th e real Hill pr ob lem. Then we exp licitly sho w that the doub le–p erio dicit y condition is equiv alen t to the existence of a real hyp erelliptic tangenti al co v er [38, 35, 36], thus completing the stud y started in [13, 1]. The conclusion is then the follo wing: the closedness p r op erty is algebraic in the parameters of the p roblem (square semiaxes of th e ellipsoid and constan ts of the motion) if and only if the double–p erio dicit y condition h olds and, in suc h a case, 2 the closedness prop ert y is equiv alen tly expressed by an elliptic in tegral asso ciated to the elliptic curv e in the h y p erelliptic tangent ial co vering. W e remark that the app earance of hyper elliptic tangen tial co vers is natural, since th eir role in th e top ologica l classification of elliptic Kd V solitons in the complex mo duli space of h yp erelliptic cur v es is we ll kno wn after T reibic h -V erdier[35]-[39] and the Moser-T rub o witz isomorphism ens ures a relation with the geo desic problem. Since the classification of closed geod esics on r eal quadrics (and of real KdV elliptic solitons) are of a certain interdisciplinary interest and the doub le-p erio dicit y prop erty of geod esics on ellipsoids is not inv ariant u nder general b irational trans- formations, we explicitly describ e suc h co v erings for the geodesic p r oblem. In par- ticular, w e inv estigate the real structur e of the elliptic curv e of the cov ering and w e sho w that the asso ciated lattice is rectangular ( i.e. all of th e ramificati on p oin ts of the elliptic curv e are r eal). W e remark also that it is appropriate to call doub ly-p erio dic the geo desics as- so ciated to hyp erelliptic tangen tial co vers, s in ce the co ordin ates and momenta are doubly–p er io dic in the length parameter s , that is they are expressed in terms of elliptic functions of s ; m oreo v er it is also appropriate to call algebraic the doubly- p erio d ic geodesics, s in ce the closedness p r op erty is algebraic in the parameters (semi- axes of the ellipsoid and constan ts of motion). The densit y prop erty The second set of questions w e characte r ize in the present pap er concerns the density c haracterization of algebraically closed geo desics. W e sho w that it is p ossible to app r o ximate a giv en real closed geod esics on a giv en ellipsoid with a sequence of real algebraically closed geo desics on p erturb ed ellip- soids with p ertu rb ed constan ts of motion. Moreo ver, su c h appro xim ate algebraically closed geo desics ma y b e c hosen so to share the same length and/or the same v alue of the p erio d ve ctor as the limiting geod esic. Our estimates are op timal in the sense that we are able to coun t the num b er of parameters which may b e ke p t fixed in the appro x im ation pro cess of real closed geod esics on a giv en n –dimens ional ellipsoid via a sequ ence of doub ly–p erio dic real closed geo desics on p erturb ed ellipsoids. F or in stance, in the simp lest case (g eo d esics on triaxial ellipsoids), Theorem 4.5 implies that we ma y k eep fixed one parameter: indeed there are four parameters (the th ree semiaxes and the caustic parameter), tw o conditions (length and p erio d mapping of the real closed geod esic to b e appro ximated algebraically) and one ex- tra condition (the approximati n g geod esics satisfy the doub le–p erio dicit y cond ition, i.e. the p erio d m ap p ing of the asso ciated imaginary geo desic has to b e rational). Similarly T heorem 4.6 implies that we ma y kee p fixed t wo parameters if we allo w the length of appro ximating algebraic geodesic to v ary a little. The densit y c haracterization follo ws from a theorem by McKean and v an Mo- erb eke [26 ] whic h allo ws the construction of a lo cally inv ertible analytic map from the s et of the parameters of the problem (the semiaxes and the caustic parameters) to the q u asi-p erio ds asso ciated to the geo desics on n –dimensional ellipsoids. 3 The case of triaxial ellipsoids: the perio d mapping and the examples W e th en sp ecialize to the case n = 2 (triaxial ellipsoids), wh ere a more detailed c haracterization of d oubly p erio dic geo desics is p ossible since the asso ciated t wo dimensional (complex) torus is isogenous to the pr o duct of t wo elliptic cu rv es. Th e first elliptic curve is asso ciated to the hyp er elliptic tangen tial cov ering, wh ile the prop erties of th e second cov ering ha v e b een discussed by Colom b o et al. [9]. In particular, we sho w that the p erio d mapp ing of a doubly p erio dic real geo desic is algebraic in the p arameters of the problem and ma y b e computed using the top ological charac ter of th e seco n d co vering. W e also work out the realit y condition for geo desics on tr iaxial ellipsoids associ- ated to degree d = 3 , 4 h yp erelliptic tangentia l co v ering and we compute the p erio d mapping usin g the top ological characte r of the asso ciated second co v ering. Finally , we pr o ve the existence of real doubly-p erio dic geod esics asso ciated to the one–parameter family of degree t wo co v er in gs with the extra automorp h ism group D 8 [17, 2]. In this case the tw o elliptic curv es of the co v ering are isomorphic, i.e. they ha ve th e same j –in v arian t, and the geod esics are d oubly-p er io dic for a dense set in the p arameter space. In view of the ab ov e discus sion, for s uc h v alues of the parameter, the giv en hyperelliptic curv e also admits another co ver w hic h is h yp erelliptic tangen tial of degree d > 2 (see Figure 4 f or an explicit example). The plan of the pap er T he plan of the pap er is th e follo wing: in the n ext section w e summarize some wel l kno wn facts ab out geo desics on n –dimensional real quadrics. In section 3, we in tro duce d oubly–p erio dic closed geo desics, h yp erelliptic tangen tial co vers and we c h aracterize the algebraic condition of closedness; in section 4 w e present the dens it y results; in sections 5 and 6 w e sp ecialize to algebraic closed geod esics on triaxial ellipsoids, we c haracterize algebraically the p erio d mappin g and we presen t th e examples. Since th e classification of closed geo desics on real quadrics and of real KdV elliptic solitons are of a certain interdisciplinary inte r est, w e h a ve tried at b est to rep ort our results in a wa y comprehen sible also to n ot exp erts in th e theory of Riemann surf aces. 2 Closed geo desics on ellipsoids The Jacobi p roblem of the geodesic m otion on an n -dimensional ellipsoid Q :  X 2 1 a 1 + · · · + X 2 n +1 a n +1 = 1  is we ll kno w n to b e in tegrable and to b e linearized on a co v ering of the Jacobian of a gen us n hyp erelliptic curve (see [27]). Namely , let l b e the n atural parameter of the geod esic and λ 1 , . . . , λ n b e the ellipsoidal co ordinates on Q defin ed b y the formulas X i = s ( a i − λ 1 ) · · · ( a i − λ n ) Q j 6 = i ( a i − a j ) , i = 1 , . . . , n + 1 . (2.1) 4 Then, denoting V i = ˙ X i = dX i /dl , i = 1 , . . . , n + 1 and ˙ λ k = dλ k /dl , k = 1 , . . . , n , the corresp onding v elo cities, the tota l energy 1 2 ( V 2 1 + · · · + V 2 n +1 ) ta kes the St¨ ac kel form H = − 1 8 n X k =1 λ k ˙ λ 2 k n Q j 6 = k ( λ k − λ j ) n +1 Q i =1 ( λ k − a i ) . According to the S t¨ a ck el theorem, the system is Liouville in tegrable. Up on fixing the constant s of motion H = h 1 , c 1 , . . . , c n − 1 and after the re-parametrizat ion dl = λ 1 · · · λ n ds √ 8 h 1 , (2.2) the evo lution of the λ k is describ ed by quadratures w hic h inv olv e n indep endent holomorphic differentia ls on a genus n hyperelliptic cur v e 1 whose affine part tak es the form Γ :  µ 2 = − λ n +1 Y i =1 ( λ − a i ) n − 1 Y k =1 ( λ − c k )  =  µ 2 = − 2 n Y i =0 ( λ − b i )  , (2.3) where we set the follo wing notation thr oughout the pap er { 0 , a 1 < · · · < a n +1 , c 1 < · · · < c n − 1 } = { b 0 = 0 < b 1 < · · · < b 2 n } . (2.4) Remark 2.1 F ollo wing [21, 3], the realit y cond ition for geo desics on ellipsoids is equiv alen t to either c i = b 2 i or c i = b 2 i +1 , i = 1 , . . . , n − 1. A first consequence is that, give n the ellipsoid Q the v alues of the r eal constan ts of motion c i s can’t take arbitrary v alues. As an example, in the s im p lest case n = 2 (triaxial ellipsoid), giv en the squ are semiaxes 0 < a 1 < a 2 < a 3 the real co n stan t of motion c s atisfies either a 1 < c < a 2 or a 2 < c < a 3 . On the other sid e, it also implies that giv en a (2 n )-tuple 0 < b 1 < · · · < b 2 n ( i.e. give n the h yp erelliptic curv e Γ), there are a finite n umber of m ec hanical con- figurations asso ciated to it. F or in stance, again in th e simplest case n = 2, to the 4-tuple b 1 < b 2 < b 3 < b 4 there are associated either the geod esics with constan t of motion c = b 2 on the ellipsoid with square semiaxes b 1 < b 3 < b 4 or the geo desics with constan t of motion c = b 3 on the ellipsoid with square semiaxes b 1 < b 2 < b 4 . Remark 2.2 Throughout the pap er, for an y give n cu rv e Γ with all real branc h p oints as in (2.3), w e use the follo win g basis of holomorphic different ials ω j = λ j − 1 dλ w , j = 1 , . . . , n , (2.5) and the homological b asis α i , β i , i = 1 , . . . , n (see Figure 1), so that the p erio ds H α i ω j ∈ R , i, j = 1 , . . . , n . 1 for th e necessary definitions and classical p rop erties of hyperelliptic curves we refer to [12, 15] 5 Then, the quadrature gi ves rise to the Ab el–Jacobi map of the n -th symmetric pro du ct Γ ( n ) to the J acobian v ariet y of Γ, P 1 Z P 0 ω j + · · · + P n Z P 0 ω j = ( s + const., for j = 1 , const., for j = 2 , . . . , n, (2.6) where and P 0 is a fi xed basep oin t an d P k = ( λ k , w k ) ∈ Γ, k = 1 , . . . , n . Then, the geod esic motion in the new p arametrization is linearized on the J acobian v a- riet y of Γ. Its complete th eta-functional solution w as p r esen ted in [40] for the case n = 2, and in [21] for arbitrary dimensions, whereas a top ological classifica- tion of r eal geod esics on qu adrics was made in [3]. In p articular, the constants of motion c 1 , . . . , c n − 1 ha ve th e follo wing geometric al meaning (see [8, 27]): the corre- sp ond ing geo d esics are tangent to the quadrics Q c 1 , . . . , Q c n − 1 of the confo cal family Q c =  X 2 1 / ( a 1 − c ) + · · · + X 2 n +1 / ( a n +1 − c ) = 1  . Figure 1. Closed geo desics and real Hill curv es Let α i , β i , i = 1 , . . . , n b e the conv en- tional homological basis depicted in Figure 1. Since we are interested in the realit y problem, it is not restrictive to take b 2 i − 1 < λ i < b 2 i , i = 1 , . . . , n , in the qu ad r atures (2.6). Then the real geodesic asso ciated to (2.6) is close d if and only if th ere exist non trivial m i ∈ Z , i = 1 , . . . , n and a real non v anishing T > 0 suc h that n X i =1 m i I α i ω 1 = T , n X i =1 m i I α i ω j = 0 , j = 2 , . . . , n, (2.7) where the basis of differentia ls h as b een int r o duced in (2.5 ). F rom (2.7), it is self- eviden t that, if a geod esic on Q is closed, then all the geo desics sh aring the same constan ts of motion c 1 , . . . , c n − 1 are closed. In the f ollo wing, we call Hill a hyp erel- liptic curve as in (2.3) for whic h (2.7 ) holds. As it is w ell kno wn , Hill curves originally arose from the stu d y of isosp ectral classes connected w ith the p erio dic Kortewe g–de V ries equ ation (see [11, 18, 24, 25, 29, 23, 26, 6]). Let H R n b e th e r eal comp onent of the mo du li space of the n on singular 6 gen us n hyp erelliptic curve s with maximal n umber n + 1 of connected components, so that all the branch p oin ts are real and distinct, b 0 = 0 < b 1 < · · · < b 2 n , then (2.7) is equiv alen t to require that Γ is a real Hill curv e up to the Moser–T rub o witz isomorphism, which is asso ciated to the birational tr ansformation z = 1 /λ (whic h exc hanges the branc h p oin ts at 0 and ∞ ). In particular, in [26], it is prov en that r eal Hill curves are dense in the mo d uli space of curves H R n . A similar statemen t holds true also for real closed geod esics; ho wev er, since the set of equations (2.7) are transcendenta l in the branch p oints of Γ, they are of little u se for the searc h of p arameters corresp ond ing to closed geod esics. F or real geo desics on ellipsoids, the ab o ve d iscussion ma y b e su mmarized in the follo w ing classical result Prop osition 2.3 F or any fixe d c hoic e of the squ ar e semiaxes a 1 < · · · < a n +1 and for any n -tuple ζ i , i = 1 , . . . , n such that a 1 ≤ ζ 1 < a 2 < · · · < a n ≤ ζ n < a n +1 , ther e is a dense set I ⊂ [ ζ 1 , ζ 2 ] × · · · × [ ζ n − 1 , ζ n ] (in the natur al top olo gy of R n − 1 ), such that ∀ ( c 1 , . . . , c n − 1 ) ∈ I , ther e exist nontrivial inte gers m i , i = 1 , . . . , n and a r e al T > 0 such that (2.7) holds. The ab o ve statemen t takes into account of the realit y condition settled in Re- mark 2.1 and exhausts all p ossibilities accordingly f or r eal closed geo desics on n - dimensional ellipsoids Q . 3 Doubly–p erio dic closed geo d esics, h yp erelliptic tan- gen tial co v ers and algebraic condition of p erio dicit y in the real parameter space In the follo wing w e consider r eal geo d esics in the r egular case when all square semi- axes and constants of m otion tak e d istinct v alues. The p erio dicit y condition (2.7 ) is transcenden tal in the parameters of the p roblem (the square semiaxes a 1 , . . . , a n of the ellipsoid and constan ts of m otion c 1 , . . . , c n − 1 ). So a natural q u estion is: is it p ossible to settle extra conditions so that the p erio dicit y condition (2.7) b ecomes algebraic in the parameters? In [13, 1], we int r o duced and c haracterized a f amily of algebraic closed geodesics asso ciated to h yp erelliptic tangen tial co v ers in the complex setting. F edoro v [13] pro ved that s uc h geodesics are a conn ected comp onent of the intersectio n of th e quadric Q with an algebraic su r face in R n . F or triaxial ellipsoids this surface is an elliptic or rational curve and the explicit d escription of the algebraic su rface in terms of elliptic P –W eierstrass functions in sp ecial cases of s uc h co v erin gs was giv en in [13]. In [1], w e computed the explicit exp ression of the co ord inates X i ( s ) in terms of one-dimensional theta-functions and applied su c h results also to d escrib e p erio dic orbits of an in tegrable b illiard. In th is section, we complete the charact erization of algebraicall y closed geod esics, w e restrict ourselv es to the real setting and we settle th e n ecessary and suffi cient con- ditions so that the closedness pr op ert y b e algebraic in the real parameters (semiaxes of the ellipsoid and constan ts of motion). 7 The conclusion is the follo win g one: the p erio d icit y condition (2.7) is alge b raic in the parameters of th e pr ob lem if and only if it is equiv alen t to the inv ersion of a single in tegral; by Jacobi in version p roblem the latter inte gral has to b e elliptic. The form of the p erio dicit y cond ition implies that the elliptic cu rv e is the one asso ciated to the hyperelliptic curv e via th e hyperelliptic tangen tial co vering. Finally , un der our h yp otheses, E has a real stru cture and w e p ro ve th at th e asso ciated lattice is rectangular (ı.e. all of the fi n ite branc h p oin ts of E are r eal). Indeed, w e in tr o duce and c haracterize a d ou b le p er io dicit y condition for the geod esics on ellipsoids w hic h is mo d elled after a similar condition for the real Hill problem by Mc Kean and v an Moer b eke [26]. Then w e explicitly show th at this condition is equiv alen t to the existence of a hyperelliptic tangen tial co ve r (explicitly describ ed in Defin ition 3.5). The theorems 3.6 [13] and 3.7 imply that the p erio dicity condition (2.7) is alge- braic in the parameters of the problem if and only if the r eal closed geo desics are doubly-p erio dic. W e remark th at it is appr opriate to call su c h geo desics doubly-p erio dic, since the co ordin ates and momenta , X i ( s ) , V i ( s ), i = 1 , . . . , n + 1, are doub ly–p erio dic in s , that is they are exp r essed in terms of elliptic fun ctions of s ; moreo ver it is also app ropriate to call algebraic the d ou b ly-p erio d ic geod esics, sin ce the closedness prop erty is algebraic in the p arameters (semiaxes of the ellipsoid and constan ts of motion). The app earance of hyp erelliptic tangen tial co v ers is natural, since their role in the top ological classificatio n of elliptic KdV solitons in the complex m o duli space of h yp erelliptic curve s is well kno wn after T reibic h -V erdier[35]-[39] and the Moser- T rub owitz isomorph ism ensures a relation with the geod esic p roblem. Since the double-p erio dicit y prop er ty of geo d esics on ellipsoids is not inv ariant under general birational transf ormations (see Lemma 3.3), we explicitly describ e suc h co v erings for the geo desic problem and we c haracterize their real structure. The plan of the section is the follo win g: we fi rst in tr o duce the double-p er io dicit y condition and c haracterize it via a dual curv e; then w e explicitly constr u ct the h yp erelliptic tangentia l co ver asso ciated to the doub le–p erio dicit y cond ition and giv e the necessary and suffi cien t conditions so th at the closedness cond ition is alg ebr aic in the parameters. Definition 3.1 (The double p erio dicity condition) A hyp er el liptic curve Γ with r e al br anch p oints as in (2.3) is asso ciate d to doubly–p erio dic close d ge o desics if and only if the r e al p erio dici ty c ondition holds, that is ther e exi sts a non trivial r e al cycle α = P n i =1 m i α i , such tha t I α ω 1 = 2 T , I α ω j = 0 , j = 2 , . . . , n , (3.1) and ther e exists a non trivial imaginary cycle β = P n i =1 m ′ i β i , such that I β ω 1 = 2 √ − 1 T ′ , I β ω j = 0 , j = 2 , . . . , n , (3.2) 8 for some non–zer o r e al T , T ′ . The conditions (3.1) and (3.2) mean that b oth the real and the imaginary geod esics on the ellipsoid Q are closed. The dual curv e T o an y giv en r eal curv e Γ like in (2.3) McKean and v an Mo erb ek e [26] asso ciate a d ual real cu r v e Γ ′ with reflected b ranc h p oints so that the real KdV elliptic soliton is doubly–p er io dic in x if and only if b oth Γ and Γ ′ are real Hill curves (for the Hill op erator). F rom the algebraic-geome tr ic p oint of view, the birational transform ation wh ic h sends branc h p oints of Γ int o those of Γ ′ is uniquely d efined by the requirement that it exc hanges real and imaginary p erio d s and it transforms h olomorph ic d ifferen tials v anishing at the infin it y ramification p oint of Γ to holomorphic differen tials v anishing at the infinity ramification p oint of Γ ′ . F rom the analytical p oin t of view, the real solutions to the Hill problem asso ciated to the dual cu r v e Γ ′ corresp ond precisely to the im aginary solutions for the corresp onding problem on Γ. W e r emark the non trivialit y of this construction: birationally equiv alent curves are identified in the mo d uli space of hyp er elliptic cur ves; h o we ver, the top ological c haracterization of the real solutions to the Hill p roblem for Kd V is not in v ariant under general bir ational transformations. The same remark h olds in the case of closed geo desics. Belo w we app ly the same idea to the case of the geod esic problem on ellipsoids: the analogous constru ction maps the imaginary geod esics on the ellipsoid Q w ith constan ts of motion c 1 , . . . , c n − 1 , to the real geo desics on a dual ellipsoid Q ′ with constan ts of m otion c ′ 1 , . . . , c ′ n − 1 . S o the r eal geo desics on Q are close d and doubly p erio d ic, if and on ly if b oth the real geo desics on Q and Q ′ are closed for the giv en constan ts of motion. T o identify the d ual curve Γ ′ ( i.e. the dual ellipsoid Q ′ and the dual constants of motion c ′ 1 , . . . , c ′ n − 1 ), we recall that the Moser-T rub o w itz isomorph ism exchanges the infinity ramification p oin t of the h yp erelliptic curv e asso ciated to the classica l Hill problem, with th e (0 , 0) fin ite ramifi cation p oint of the hyp erelliptic curve asso ciated to the geo d esic problem. Moreo v er, either the r eal (3.1) or the imaginary (3.2) p erio dicit y conditions for the geo desic problem are equiv alen t to require that, giv en a hyper elliptic curve as in (2.3), there exists a non tr ivial cycle γ such that H γ ω = 0, f or all holomorphic differen tials ω v anishing at the branc h p oin t (0 , 0). Lemma 3.2 L et Γ : { µ 2 = − 2 n Q k =0 ( λ − b k ) } b e as in (2.3), let P 0 = (0 , 0) and let ω k = λ k − 1 /µ , k = 1 , . . . , n , b e the b asis of holomor phic differ entials intr o duc e d i n (2.5). Then the holomor phic differ ential ω v anishes at the br anch p oint P 0 if and only if it is a line ar c ombination of the holomorphic differ entials ω 2 , . . . , ω n . Sketch of the pr o of: Let τ b e the lo cal co ordin ate in a neigh b orho o d of P 0 = (0 , 0) such that τ ( P 0 ) = 0, then ω k ≈ Aτ 2 k − 2 dτ , k = 1 , . . . , n , where A = 9 2  q − Q 2 n j =1 b j  − 1  . Finally , to construct the dual curve Γ ′ for the geo desic prob lem, w e must iden tify the birational transformations whic h preserve the form of (2.3) and transform holo- morphic differenti als v anishing at th e ramification p oint (0 , 0) ∈ Γ to h olomorph ic differen tials v anishing at the ramification p oin t (0 , 0) ∈ Γ ′ . Lemma 3.3 The class of bir ational tr ansform ations b etwe en Γ : { µ 2 = − 2 n Q k =0 ( λ − b k ) } and Γ ′ : { ν 2 = − 2 n Q k =0 ( ρ − b ′ k ) } which tr ansform holomorp hic differ entials van- ishing at the r amific ation p oint (0 , 0) ∈ Γ to holomorphic differ entials vanishing at the r amific ation p oint (0 , 0) ∈ Γ ′ has the fol lowing two gener ators: ρ = κλ and ρ = b 1 λ/ ( λ − b 1 ) . The first transformation is a homogeneous rescaling of all of the parameters of the p r oblem (squ are semiaxes and constan ts of the motion) and it preserve s the real p erio d icit y condition. Th e latter transformation is the analog of the one intro d uced b y McKea n and v an Mo erb ek e [26 ] for the p erio dic Kd V problem and it exc hanges the real and the imaginary cycles. Finally , the statemen t b elo w giv es a simple c h aracterization of doubly–p erio dic geod esics and is the analog of a theorem in [26] for th e Hill p roblem. Theorem 3.4 L et Γ = { µ 2 = − λ 2 n Q k =1 ( λ − b k ) ≡ − λ n +1 Q i =1 ( λ − a i ) n − 1 Q k =1 ( λ − c k ) } b e a r e al hyp er el liptic curve as in (2.3) and Γ ′ : { ν 2 = − 2 n Q i =0 ( ρ − b ′ i ) ≡ − λ n +1 Q i =1 ( λ − a ′ i ) n − 1 Q k =1 ( λ − c ′ k ) } b e the r e al hyp er el liptic curve whose br anch p oints b ′ i ar e r elate d to the b j s by the bir ational tr ansformation ρ = b 1 λ λ − b 1 . L et Q = { X 2 1 /a 1 + · · · + X 2 n +1 /a n +1 = 1 } and Q ′ = { X 2 1 /a ′ 1 + · · · + X 2 n +1 /a ′ n +1 = 1 } . Then the r e al ge o desics asso ciate d on Q with c onstants of motion c 1 , . . . , c n − 1 ar e doubly–p erio dic if and only if the r e al ge o desics r e sp e ctively asso ciate d to Q (with c onstant s of motion c 1 , . . . , c n − 1 ) and to Q ′ (with c onstants of motion c ′ 1 , . . . , c ′ n − 1 ) ar e close d. In view of Remark 2.1, to any giv en (2 n )-tuple b ′ 1 , . . . , b ′ 2 n , there are asso ciated a finite n umb er of du al ellipsoids and du al constan ts of motion. Clearly th e Theorem 3.4 imp lies th at the real geodesics on Q ′ b e closed, for an y admissib le dual ellipsoid Q ′ and constan ts of motion c ′ 1 , . . . , c ′ n − 1 asso ciated to b ′ 1 , . . . , b ′ 2 n . In the last sect ion, w e apply Th eorem 3.4 b oth to compu te the p erio d mapping asso ciated to families of co verings and to compute the parameters of Examp le 6.3. 10 Hyp erelliptic tangential co vers and the algebraic p erio dicit y condition for closed geo desics In this paragraph, we pr o v e that the double–p erio dicit y condition is necessary and sufficient for the algebraic c h aracterizati on of the closed- ness prop ert y of real geod esics on n –dimensional ellipsoids. The statemen t follo ws from the f act th at th e double p erio dicit y condition settled by equations (3.1) and (3.2) is equiv alen t to th e existence of a real rectangular hyperelliptic tangen tial co ver defined in Definition 3.5. Hyp erelliptic tangen tial co ve r s [35]-[38] ha ve originally app eared in conn ection with the top ologica l classification of the x doubly–p er io dic solutions of the Kortewe g- de V ries (KdV) equ ation u t = 6 uu x − u xxx . Due to the Moser–T rub o w itz isomor- phism, we get a natural relation b et ween the classification of real d oubly–p erio d ic geod esics and the relev ant class of p erio dic p oten tials asso ciated to th e Hill op erator − ∂ 2 x + u ( x, t ), dep ending on the p arameter t (due to th e imp ossibilit y of citing all relev ant con tr ibutions in this field we limit to cite [11, 10, 18, 24, 25, 4]). W e recall that a solution to the Kd V equation of the form u ( x, t ) = 2 N P j =1 P ( x − q i ( t )) + c is called a K d V–elliptic s oliton. u ( x, t ) is a KdV-elliptic soliton if and only if P 1 ≤ j ≤ N ,j 6 = k P ′ ( q j ( t ) − q k ( t )) = 0, k = 1 , . . . , N [4]. An y KdV–elliptic soliton is un iquely asso ciated with a mark ed h yp er elliptic curv e ( X, P ) of p ositiv e gen us g equipp ed with a p r o jection π : X 7→ E the so cal led h yp erelliptic tangen tial co ve r - suc h that P is a smo oth W eierstrass p oin t of X and the canonical images of ( X, P ) and ( E , Q ) in the Jacobian of X are tangen t at the origin [36]. The problem of classifying all hyp erelliptic tangentia l cov ers in the complex mo duli sp ace of genus g h yp erelliptic curves and to c haracterize the asso ciated Kd V– elliptic solitons has b een successfully considered in a series of p ap ers b y T reibich and V erdier [35]-[39]. W e refer to [38] f or an accoun t of the v ast literature on the sub ject. In particular, a different appr oac h to the classification problem of KdV–elliptic solitons has b een dev elop ed by K ric hever [23] based on th e theory of one p oin t Bak er–Akhiezer functions, wh ile Gesztesy and W eik ard [14] give an analytic c h ar- acterizat ion of elliptic finite–gap p oten tials. Finally , explicit examples of families of suc h co ve r ings hav e b een work ed out by many authors (see in particular [32, 35, 33]). In [1, 13], hyp erelliptic tangentia l cov ers w ere fir st considered in connection to doubly–p er io dic closed geo desics on n –dim en sional (complex) quadrics and explicit examples w ere w orked out. In particular, a theorem by F edoro v[13 ] implies if the curv e (Γ , P 0 ) is a (complex) hyp erelliptic tangen tial co ve r , th en th e geo desics on the asso ciated qu adric are (complex) d oubly p erio dic. Here we restrict ourselves to real hyp erelliptic curves Γ with all finite b ranc h p oints real. F or su c h curv es w e call the hyperelliptic ta n gen tial co v erin g real (resp. real rectangular, real rh om bic), if the elliptic curve E has a real stru cture (resp. w ith rectangular, r h om bic p erio d lattice). The theorem b y F edoro v ma y b e easily r ephrased so to hold in the case of r eal tangen tial co verings. Moreo ver, here we prov e the rev erse statemen t: if the d ouble– p erio d icit y condition (3.1) and (3.2) hold, then the asso ciated algebraic cu rv e is a 11 real hyperelliptic tangen tial co ver. Finally , in th e latter case we sho w th at it is alw a ys p ossible to asso ciate to the hyp erelliptic curv e for whic h the d ouble p er io dicit y condition holds, a real rectangular hyp erelliptic tangen tial co v erin g. The conclusion is then that the double–p er io dicit y condition is necessary and suf- ficien t for the algebraic c h aracterizatio n of th e closedness pr op ert y of real geo desics on n –dimensional elli p soids. Definition 3.5 Real rectangular h yp erelliptic t angen tial co verings L et Γ : { µ 2 = − 2 n Q k =0 ( λ − b k ) } b e as in (2.3), let P 0 = (0 , 0) and let ω k = λ k − 1 /µdλ , k = 1 , . . . , n , b e the b asis of holomorphic differ entials intr o duc e d in (2.5). L et A = 2  q − Q 2 n j =1 b j  − 1 b e as in the pr o of of L emma 3.2. The curve Γ admits a c anonic al emb e dding i nto its Jac obian variety Jac(Γ) by the map P 7→ A ( P ) = R P P 0 ( ω 1 , . . . , ω n ) T , so that P 0 is mapp e d into the origin of the Jac obian and U = d dτ A ( P )   P = P 0 = ( A, 0 , . . . , 0) , i s the tangent ve ctor of Γ ⊂ Jac(Γ) at the origin. Assume that Γ is an N –fold c overing of an el liptic curve E , which we r epr esent in the c anonic al Weierstr ass form E = { ( P ′ ( u )) 2 = 4 P 3 ( u ) − g 2 P ( u ) − g 3 ≡ 4( P ( u ) − e 1 )( P ( u ) − e 2 )( P ( u ) − e 3 ) } . Assume that under the c overing map π : Γ 7→ E , P 0 is mapp e d to Q 0 the infinite p oint of E and cho ose u as lo c al c o or dinate. The c overing fr om the marke d curve (Γ , P 0 ) to ( E , Q 0 ) is hyp er el liptic al ly tan- gential if E admits the fol lowing c anonic al emb e dding to Jac(Γ) , u 7→ u U , so that the emb e dding of Γ and E ar e tangent at the origin. We c al l (Γ , P 0 ) a real h yp erelliptic t a ngen tial cov ering if the ab ove holds and the el liptic cu rve E has a r e al structur e (i.e. the p erio d lattic e asso ciate d to E is either r e ctangular or rhombic). We c al l the r e al hyp er el liptic tangential c overing (Γ , P 0 ) rectangular i f mor e- over al l the finite br anch p oints of E ar e r e al (so the lattic e asso ciate d to E is r e ct- angular). Otherwise, we c al l the r e al hyp er el liptic tangential c overing rhom b ic. F or th e geo desic problem, the existence of a real h yp erelliptic tangentia l co ve r ing implies the d ouble-p erio d icit y cond ition b y the follo wing theorem. Theorem 3.6 If (Γ , P 0 ) is a r e al hyp er el liptic tangential c over, then the asso ciate d ge o desics ar e close d and doubly–p erio dic. The ab o ve theorem was originally pro ven by F edoro v [13] in the complex setting: indeed if (Γ , P 0 ) is a hyp erelliptic tangent ial co ver, then th e complex geo desics on the quadric Q satisfy a d ouble–p erio d icit y condition. His argument ma y b e easily mo dified so to hold in the real setting. W e remark that w e get th e doub le-p erio dicit y condition (3.1)-(3.2) either if the real h yp erelliptic tangen tial co v ering is rectangular or rhombic. 12 The ab o ve th eorem settles a sufficient condition for the algebraicit y of the closed- ness p r op erty of real geo desics on ellipsoids. Next theorem implies that such condi- tion is also necessary; so that we get the complete charac terization of algebraically closed geo desics via the double–p er io dicit y condition. W e now pr o ve th e con v erse to Th eorem 3.6. Theorem 3.7 L et Γ = { µ 2 = − λ n +1 Q i =1 ( λ − a i ) n − 1 Q k =1 ( λ − c k ) ≡ − λ 2 n Q k =1 ( λ − b k ) } b e the hyp er el liptic curve asso ciate d to the ge o desics on the el lipsoid Q = { X 2 1 /a 1 + · · · + X 2 n +1 /a n +1 = 1 } with c onstants of motion c 1 , . . . , c n − 1 . L et α 1 , . . . , α n , β 1 , . . . , β n b e the c onventional c anonic al homolo gic al b asis depicte d in Figur e 1 and let ω j , j = 1 , . . . , n b e the b asis of holomorp hic differ e ntials intr o duc e d in (2.5). If the doubly–p erio dicity c onditions (3.1) and (3.2) hold, then (Γ , P 0 ) is a r e al r e ctangular hyp er el liptic tangential c over. Pr o of of The or em 3.7 The doubly–p erio dicit y conditions (3.1) and (3. 2 ) hold if and only if there exist t wo cycles α = n P i =1 m i α i and β = n P i =1 m ′ i β i , such that I α ω j =  T , j = 1 , 0 , j = 2 , . . . , n, I β ω j =  √ − 1 T ′ , j = 1 , 0 , j = 2 , . . . , n. (3.3) The ab o v e equations imply that ω 2 , . . . , ω n are the ( n − 1) indep endent holomorph ic differen tials v anishin g at P 0 = (0 , 0) and p ossess a maximal system of (2 n − 2) inde- p end ent p erio d s. Th en by P oincar ´ e reducibilit y theorem [31], there exist an elliptic curv e E and a ( n − 1)–dimensional Ab elian subv ariet y A n − 1 suc h that Jac(Γ) is isogenous to the d ir ect pro duct E × A n − 1 . S ince P 0 = (0 , 0) is among the W eier- strass p oin ts of Γ, the co vering π : Γ 7→ E is tangen t at the W eierstrass p oin t P 0 [39, 36]. Since all of the W ei er s trass p oints of the curve Γ are real (see 2.3) and since th e double p erio dicit y condition (3.3) en sures the rational d ep endence b et ween the r eal p erio d s (asso ciated to the α cycle) and the rational dep endence b et ween the imagi- nary p erio d s (associated to β ), we easily conclud e that the h yp erelliptic tangen tial co v ering has a real structure. W e no w explicitly constru ct su c h co vering in order to inv estiga te the real struc- ture asso ciated to E . The tangency condition and the (3.3) ensure th e existence of t wo real n u m b ers A, B , of a h olomorphic differen tial Ω 1 = ω 1 + P n j =2 c j ω j , and of constan ts k 1 . . . , k n , h 1 , . . . , h n ∈ Z , su c h that I α j Ω 1 = 2 k j A, I β j Ω 1 = 2 h j √ − 1 B , j = 1 , . . . , n. Since α 1 , . . . , α n , β 1 , . . . , β n form a homological basis, any other p erio d of Ω 1 is an in teger com bination of 2 A and 2 √ − 1 B . In p articular, T = I α Ω 1 = A n X j =1 k j m j , √ − 1 T ′ = I β Ω 1 = √ − 1 B n X j =1 h j m ′ j . 13 W e now inv estigat e the real structure of the co vering. Let us fix P 0 = (0 , 0) ∈ Γ as basep oint, let z = R P P 0 Ω 1 , P ∈ Γ. Then z ∈ T = C / Λ, the one–dimensional torus with p erio d lattice Λ generated b y 2 A, 2 √ − 1 B . Finally let P ( z ) ≡ P ( z | A, √ − 1 B ) b e the W eierstrass P -fu nction with half- p erio d s A, √ − 1 B and E :   P ′ ( z )  2 = 4 3 Q k =1  P ( z ) − e k   the elliptic curve in W eierstrass normal form with finite b ranc h p oint s e 1 = P ( A ), e 2 = P ( A + √ − 1 B ) and e 3 = P ( √ − 1 B ). Then, the co v erin g π : Γ 7→ E is real rectangular and tangen tial at P 0 = (0 , 0) b y co n struction.  W e remark that there is a certain freedom in the constru ction of the curve E and of the co vering, due to the isogeneit y b et ween Jac(Γ) and E × A n − 1 . F or instance, if w e in tro d uce the complex conju gate num b ers C ± = A ± √ − 1 B , we ma y asso ciate to (Γ , P 0 ) a real rhom bic hyp erelliptic tangen tial co ve r ing. Theorem 3.7 means that the d ouble-p erio d icit y condition is algebraic in the parameters of the problem (the square semiaxes a 1 , . . . , a n +1 and the constan ts of motion c 1 , . . . , c n − 1 ), sin ce it may b e equiv alen tly expressed in terms of elliptic in tegrals associated to the co ve r ing. Theorem 3.6 means that for the sp ecial class of geo desics on ellipsoids asso ciated to a r eal h yp erelliptic tangen tial co v ering (Γ , P 0 ), the p erio dicit y condition (3.1) n X i =1 m i I α i ω 1 = 2 T , n X i =1 m i I α i ω j = 0 , j = 2 , . . . , n, is algebraic in the p arameters of the pr ob lem, since the co v erin g π imp oses algebraic relations among the branch p oin ts of E and the ramifi cations p oin ts of Γ (squ are semiaxes a 1 , . . . , a n +1 and constan ts of motion c 1 , . . . , c n − 1 ), and the real (resp. imaginary) p erio dicity cond ition is expressible as a real (resp . imaginary) elliptic in tegral on E . W e thus get the follo wing Corollary 3.8 L et Γ = { µ 2 = − λ n +1 Q i =1 ( λ − a i ) n − 1 Q k =1 ( λ − c k ) ≡ − λ 2 n Q k =1 ( λ − b k ) } b e the hyp er el liptic curve asso ciate d to the ge o desics on the el lipsoid Q = { X 2 1 /a 1 + · · · + X 2 n +1 /a n +1 = 1 } with c onstants of motion c 1 , . . . , c n − 1 . L et α 1 , . . . , α n , β 1 , . . . , β n b e the c onventional c anonic al homolo gic al b asis depicte d in Figu r e 1 and let ω j = λ j − 1 dλ/µ , j = 1 , . . . , n b e the b asis of holomorphic differ entials intr o duc e d in (2.5). L et P 0 = (0 , 0) ∈ Γ . Then the close dness pr op erty (2.7) n X i =1 m i I α i ω 1 = 2 T , n X i =1 m i I α i ω j = 0 , j = 2 , . . . , n , is algebr aic in the p ar ameters of the pr oblem a 1 . . . , a n +1 , c 1 , . . . , c n − 1 (squar e semi- axes and c onstant s of motion), if and only if ther e exists a non trivial imaginary 14 cycle β = P n i =1 m ′ i β i , such tha t I β ω 1 = 2 √ − 1 T ′ , I β ω j = 0 , j = 2 , . . . , n . In the latter c ase, (Γ , P 0 ) is a r e al r e ctangular hyp er el liptic tangential c over. The Corollary is p erf ectly consistent with the T reibic h–V erdier c haracterizat ion of elliptic solitons of the Kortew eg–de V ries equ ations (we refer in particular to [39] for a discussion of the dimension of the real mo d uli space asso ciated to either the p erio d ic or double–p erio dic stationary solution to the KdV equ ation). On the other sid e the Corollary implies that the p erio dicity condition for the geod esic problem will sta y transcendental for an y o ve r t yp e of co v erin g: for ins tance the p erio dicity condition will sta y transcendenta l, if Γ as in (2.3) is a hyp erelliptic tangen tial co ver with mark ed p oint P j = ( b j , 0), for some j = 1 , . . . , 2 n or if Γ is a degree d = 2 co vering (the d egree of a hyp erelliptic tangen tial co ver is at least 3). In particular, in the last section w e pro ve the existence of doubly–p erio dic closed geod esics related to degree 2 cov erings with extra automorphisms and we give an explicit example (see Figure 4): in view of Corollary 3.8 in su c h case the cur v e admits also a h yp erelliptic tange ntial co v er, and then an infin ite num b er of co verings b y a classical theorem by Picard [30]. 4 Densit y of doubly–p erio dic closed geo d esics In this section, w e p ro ve that the algebraic condition of r eal closed geo desics settled in th e p r evious section, is fulfilled on a dense set of parameters (the square semiaxes a 1 , . . . , a n +1 and the constan ts of m otion c 1 , . . . , c n − 1 ) with r esp ect to th e natur al top ology ov er the r eals. So it is p ossible to c h aracterize algebraically dense sets of real closed geo desics on ellipsoids and to approxima te real closed geo desics on giv en ellipsoid b y sequ en ces of algebraicall y closed (i.e. doubly-p erio dic) geodesics on p ertu rb ed ellipsoids w ith p er tu rb ed constan ts of motion. W e remark that, su c h app ro ximate algebraically closed geod esics ma y b e c hosen so to share the same length and/or the same v alue of the p erio d v ector as the limiting geod esic. Our estimates are op timal in the sense that w e are able to count the n u m b er of parameters wh ic h ma y b e kept fixed in this app ro ximation sc heme. F or instance, in the simplest case (ge o desics on triaxia l ellipsoids), T h eorem 4.5 implies that w e ma y kee p fixed one parameter: indeed we ha ve four p arameters (the three s emi- axes and the caustic parameter), t wo conditions originating from the limiting closed geod esics (length T and p erio d mapping m 1 /m 2 ) and one extra condition (the ap- pro ximating geo desics hav e rational v alue of the imaginary p erio d mapping m ′ 1 /m ′ 2 whic h appro ximates the irrational quasi–p erio d of the limiting imaginary geo desic). Similarly Theorem 4.6 implies th at w e ma y k eep fixed tw o p arameters (since we also p ertur b the length of the appro ximating algebraic geodesics). 15 The pro ofs of th e d ensit y results rely on a theorem by McKean and v an Mo- erb eke for the Hill pr oblem [26]. Using their idea, we d efine a qu asi-p erio d ve c- tor ( x, y ) ≡ ( x 1 , . . . , x n , y 1 , . . . , y n ) ∈ R 2 n asso ciated to any real and imaginary geod esics. Using the Riemann bilinear relations, such quasi–p erio d vec tor ma y b e explicitly computed usin g the p erio ds of t wo meromorphic differen tials. The theo- rem by [26] (originally stated for the Hill problem), ens ures that the map f rom the parameter sp ace ( a 1 , . . . , a n +1 , c 1 , . . . , c n − 1 ) to the qu asi p erio d s ( x, y ) is analytic and lo cally in v ertible. Densit y of a lgebraically closed geo desics F o r an easie r comparison with the densit y c haracterization of KdV-elliptic solitons, we also rep ort the follo w ing c har- acterizat ion of h yp erelliptic tangen tial cov ers in the complex mo d uli space of hyper- elliptic cur v es due to Colombo et al. [9]. Th eir theorem imp lies immediately th at real closed geo desics ma y b e appro ximated b y co m plex d ou b ly–p erio d ic geo desics. Theorem 4.1 [9] Hyp er el liptic tangential c overs of g e nus n ar e dense in the c om- plex mo duli sp ac e H n of the hyp er el liptic curve s of genus n . T o p ro ve th e density statemen t (Theorem 4.3) for real doubly p erio dic geo desics on n –dim en sional ellipsoids with resp ect to the real parameter space, we apply th e ideas used by McKean and V anMoerb ek e in [26] for the Hill problem. W e rep ort their theorem b elo w in a v ersion suitable for the geod esics problem and then sho w that an y real closed geodesics on a giv en ellipsoid ma y b e appr o ximated by r eal doubly–p er io dic geodesics on p er tu rb ed ellipsoids. Theorem 4.2 [26] L et Γ = { µ 2 = − λ n +1 Q i =1 ( λ − a i ) n − 1 Q k =1 ( λ − c k ) ≡ − λ 2 n Q k =1 ( λ − b k ) } b e as in (2.3). L et ( x, y ) = ( x 1 , . . . , x n , y 1 , . . . , y n ) ∈ R 2 n b e define d by n X i =1 x i I α i ω j =  1 , for i = 1 , 0 , for i = 2 , . . . , n, n X i =1 y i I β i ω j =  √ − 1 , for i = 1 , 0 , for i = 2 , . . . , n . (4.1) Then, (4.1) define a r e al analytic lo c al ly invertible map fr om op en sets in the p ar am- eter sp ac e ( b 1 , . . . , b 2 n ) to op en sets in the quasi–p erio d sp ac e ( x, y ) = ( x 1 , . . . , x n , y 1 , . . . , y n ) . In p articular, a smal l p erturb ation of the r e al br anch p oints of Γ wil l make the p oint ( x, y ) r ational. If we compare (4.1 ) with the double-p erio dicit y condition settled in (3.1 ) and (3.2), n X i =1 m i I α i ω j =  T , for i = 1 , 0 , for i = 2 , . . . , n, n X i =1 m ′ i I β i ω j =  √ − 1 T ′ , for i = 1 , 0 , for i = 2 , . . . , n , 16 w e easily conclude that if the p oint ( x, y ) is rational, then th e double p erio d icit y condition is satisfied. Then the follo wing densit y pr op ert y of algebraically closed geod esics holds. Theorem 4.3 Given a r e al close d ge o desic on the e l lipsoid Q = { X 2 1 /a 1 + · · · + X 2 n +1 /a n +1 = 1 } with c austic p ar ameters c j , j = 1 , . . . , n − 1 , for any ǫ > 0 suffi- ciently smal l, ther e exist a ǫ 1 , . . . , a ǫ n +1 , c ǫ 1 , . . . , c ǫ n − 1 ∈ R such that n − 1 X j =1 ( c j − c ǫ j ) 2 + n +1 X i =1 ( a i − a ǫ i ) 2 < ǫ and the ge o desics on Q ǫ = { X 2 1 /a ǫ 1 + · · · + X 2 ( n +1) /a ǫ n +1 = 1 } with c austic p ar ameters c ǫ j , j = 1 , . . . , n − 1 , ar e r e al doubly p erio dic. Pr o of: L et Γ b e th e real Hill curv e asso ciated to the closed geo desics on the ellipsoid Q with caustic p arameters c 1 , . . . , c n − 1 so that the set of equations (3.1) hold. Let ǫ 0 = 1 2 min { b j − b j − 1 , j = 1 , . . . , 2 n } where { b 1 < · · · < b 2 n } = { a 1 < · · · < a n +1 , c 1 < · · · , c n − 1 } . Γ ′ , the dual curve to Γ in tro duced in Theorem , is asso ciated to a dual ellipsoid Q ′ whic h p ossesses r eal quasi–p erio dic closed geod esics, so that the vect or y = ( y 1 , . . . , y n ) ∈ R n . Similarly to [26], we introdu ce the differen tial of the second kind Ω (0) 2 with v an- ishing β j p erio d s, w ith a double p ole at P 0 = (0 , 0) and the follo wing normalization. Let τ b e th e lo cal co ordinate in a neigh b orh o o d of P 0 = (0 , 0) suc h that τ ( P 0 ) = 0, then Ω (0) 2 ≈ (2 π A ) − 1 τ − 2 dτ , k = 1 , . . . , n , w h ere A = 2  q − Q 2 n j =1 b j  − 1 is the constan t defined in Definition 3.5. L et y j = I α j Ω (0) 2 , j = 1 , . . . , n , b e the α –p erio d vecto r of Ω (0) 2 . Then applying Riemann bi–linear id en tities to ω l , l = 2 , . . . , n and to ˜ u = R P P 0 Ω (0) 2 , we immediately conclude that n X j =1 y j I β j ω 1 = √ − 1 , n X j =1 y j I β j ω l = 0 , l = 2 , . . . , n. Finally , app lying Theorem 4.2, we m a y p erturb th e curve Γ so that on Γ ǫ (with ǫ < ǫ 0 ) n X j =1 x ( ǫ ) j I α ( ǫ ) j ω ( ǫ ) 1 = 1 , n X j =1 y ( ǫ ) j I β ( ǫ ) j ω ( ǫ ) 1 = √ − 1 , n X j =1 x ( ǫ ) j I α ( ǫ ) j ω ( ǫ ) l = 0 , n X j =1 y ( ǫ ) j I β ( ǫ ) j ω ( ǫ ) l = 0 , l = 2 , . . . , n, for rational v ector ( x ( ǫ ) , y ( ǫ ) ). According to Theorem 3.7 (Γ ( ǫ ) , P 0 ) is a hyp erelliptic tangen tial co v er.  17 Remark 4.4 It is easy to v erif y that the v ectors ( x, y ) in (4.1) corresp ond to a h yp erelliptic tangen tial co ver if and only if x = ( x 1 , . . . , x n ) has r ationally dep endent comp onent s and the same holds for y = ( y 1 , . . . , y n ) (that is the requirement that ( x, y ) b e rational m a y b e wea kened, w ithout lo osing the algebricit y of the closedness condition of the asso ciated geo desics) . In view of th e ab ov e remark, it is p ossible to optimize the d ensit y c haracterization of doubly p erio dic closed geodesics. Ind eed it is p ossible to mo dify the pro of of the ab o ve theorem so that th e ellipsoids Q , Q ǫ share the same v alue of the greatest squ are semiaxis a n +1 = a ǫ n +1 , and the p erturb ed real dou b ly-p erio d ic closed geod esics on Q ǫ ha ve the s ame length and the same p erio d v ector as the initial real closed geo desics on Q , i . e. ( x 1 , . . . , x n ) = ( x ǫ 1 , . . . , x ǫ n ). Theorem 4.5 L et Γ = { µ 2 = − λ ( λ − c ) n +1 Q i =1 ( λ − a i ) n − 1 Q j =1 ( λ − c j ) } , b e a r e al Hil l curve so that the r e al ge o desics on the el lipsoid Q = { X 2 1 /a 1 + · · · + X 2 n +1 /a n +1 = 1 } with c austic p ar ameters c 1 , . . . , c n − 1 ar e close d and have length T . Then, ther e exists a se quenc e { a ( k ) 1 , . . . , a ( k ) n , c ( k ) 1 , . . . , c ( k ) n − 1 } ∈ R 2 n − 1 such that lim k → + ∞ c ( k ) j = c j , ( j = 1 , . . . , n − 1) , lim k → + ∞ a ( k ) i = a i , ( i = 1 , . . . , n ) , and the ge o desics on Q ( k ) = { X 2 1 /a ( k ) 1 + · · · + X 2 n /a ( k ) n + X 2 n +1 /a n +1 = 1 } with c austic p ar ameters c ( k ) = ( c ( k ) 1 , . . . , c ( k ) n − 1 ) ar e doubly–p erio dic, with same length T and with the same value of the p erio d ve ctor as the close d ge o desics on ( Q, c 1 , . . . , c n − 1 ) . Pr o of: The pro of follo w s from a straigh tforw ard adap tation of the argumen t in Theorem 4.2 : since the jacobian determinan t of the real analytic map there defin ed is not v anishing, also its restriction to a generic 2 n − 1–dimensional subv ariet y will not v anish lo cally . T o fix ideas, we c ho ose th e subv ariety b 2 n ≡ a n +1 = const. . Let Γ b e real Hill, let ω 1 , . . . , ω n b e the h olomorphic b asis of differen tials defined in (2.5) and α i , β i , i = 1 , . . . , n the h omologica l basis as in Remark 2.2. Let Ω (0) 2 b e the normalized meromorphic differen tial of the second kind with double p ole at P 0 = (0 , 0), v anishin g β p erio ds, as in the pr o of of Theorem 4.3 , an d let ( y 1 , . . . , y n ) b e its α p erio d vec tor. Let ǫ 0 = 1 2 min { b j − b j − 1 , j = 1 , . . . , 2 n } , wh ere, as u sual { b 1 < · · · < b 2 n } = { a 1 , . . . , a n +1 , c 1 , . . . , c n − 1 } . Then the geo desics on Q with caustic p arameters c 1 , . . . , c n − 1 are real closed and satisfy the p erio dicit y condition f 1 ( b 1 , . . . , b 2 n ) ≡ n X i =1 m i I α i ω 1 − T = 0 , f j ( b 1 , . . . , b 2 n ) ≡ n X i =1 m i I α i ω j = 0 , j = 2 , . . . , n. (4.2) 18 Let b 2 n , m 1 , . . . , m n , T b e fixed. As a consequence of Theorem 4.2, the n equatio n s f j = 0, j = 1 , . . . , n are lo cally analytical ly inv ertible near the p oin t ( b 1 , . . . , b 2 n − 1 ) and there exist n analytic functions ˆ b r = ˆ b r ( ˆ b 1 , . . . , ˆ b n − 1 ), r = n, . . . , 2 n − 1, on the ( n − 1)–dimensional ball B 0 cen tered at ( b 1 , . . . , b n − 1 ) and of rad iu s ǫ < ǫ 0 . On the initial curv e Γ, g j ≡ y j /y n = Z b 2 j b 2 j − 1 Ω (0) 2 / Z b 2 n b 2 n − 1 Ω (0) 2 , j = 1 , . . . , n − 1 tak e some r eal v alue τ j , j = 1 , . . . , n − 1 and are real analytic in ˆ b 1 , . . . , ˆ b n − 1 on the ball B 0 , again by Theorem 4.2. Then, there exists a sequence ( b ( k ) 1 , . . . , b ( k ) n − 1 ) ∈ B 0 con ve r ging to ( b 1 , . . . b n − 1 ) suc h that lim k → + ∞ b ( k ) r ≡ lim k → + ∞ ˆ b r ( b ( k ) 1 , . . . , b ( k ) n − 1 ) = b r , r = n, . . . , 2 n − 1; g j ( b ( k ) 1 , . . . , b ( k ) n − 1 ) ∈ Q , j = 1 , . . . , n − 1; lim k → + ∞ g j ( b ( k ) 1 , . . . , b ( k ) n − 1 ) = τ j , j = 1 , . . . , n − 1 . (4.3) Finally , for an y k , b y construction, the corresp onding hyp erelliptic cur v e Γ ( k ) = { µ 2 = − λ 2 n Q j =1 ( λ − b ( k ) j ) } is a hyp erelliptic tangen tial cov er with marke d p oint P 0 = (0 , 0) and the asso ciated geo desics ha ve th e same length and the same p e- rio d v ector as the initial ones asso ciated to Γ. Ind eed equations (4.2) ensure th at on Γ ( k ) the p erio d v ector and the length of the real geodesics b e preserve d ; b y (4.3), the imaginary p erio d v ector ( y ( k ) 1 , . . . , y ( k ) n ) has rationally dep endent comp onents for all k w hic h, appro ximate the rationally indep endent components of the imagi- nary qu asi–p erio d of the limiting imaginary geo desics, so that, by constru ction, the limiting real closed geod esics are th ose asso ciated to Γ.  Finally , if we ju s t r equire to p reserv e the p erio d ve ctor of the geodesics and allo w that the length of the app ro ximating geod esics v ary , ı.e. if w e just r equire ( x 2 /x 1 , . . . , x n /x 1 ) = ( x ǫ 2 /x ǫ 1 , . . . , x ǫ n /x ǫ 1 ), w e ma y k eep fixed tw o s quare semiaxes, for instance the smallest and the greatest one, a 1 = a ǫ 1 and a n +1 = a ǫ n +1 and we get the follo win g statemen t. Theorem 4.6 L et Γ = { µ 2 = − λ ( λ − c ) n +1 Q i =1 ( λ − a i ) n − 1 Q j =1 ( λ − c j ) } , b e a r e al Hil l curve so that the r e al ge o desics on the el lipsoid Q = { X 2 1 /a 1 + · · · + X 2 n +1 /a n +1 = 1 } with c austic p ar ameters c 1 , . . . , c n − 1 ar e close d and have length T . Then, ther e exists a se quenc e { a ( k ) 2 , . . . , a ( k ) n , c ( k ) 1 , . . . , c ( k ) n − 1 } ∈ R 2 n − 2 such that lim k → + ∞ c ( k ) j = c j , ( j = 1 , . . . , n − 1) , lim k → + ∞ a ( k ) i = a i , ( i = 2 , . . . , n ) , 19 and the ge o desics on Q ( k ) = { X 2 1 /a 1 + X 2 2 /a ( k ) 2 + · · · + X 2 n /a ( k ) n + X 2 n +1 /a n +1 = 1 } with c austic p ar ameters c ( k ) = ( c ( k ) 1 , . . . , c ( k ) n − 1 ) ar e doubly–p erio dic, with same value of the p erio d ve ctor as the c lose d ge o desics on ( Q, c 1 , . . . , c n ) . The pro of is a straightfo r w ard mo d ifi cation of the one for Theorem 4.5 and we omit it. Remark In [1], w e u sed th e algebraic charac terization of closed geo desics associ- ated to h yp erelliptic tangen tial cov ers to constru ct p erio dic billiard tra jectories of an inte grab le billiard on a qu adric Q with elastic imp acts on a confocal quadric Q d . The results w e h a ve presente d in this sectio n may b e applied to this b illiard mo del and imply th e algebraic c h aracterizatio n of a dense set of its p erio dic orbits. 5 The algebraic computation of the p erio d mapping in the case n = 2 In the sp ecial case of triaxial ellipsoids ( n = 2), a stronger characte r ization of doubly-p erio dic closed geo desics h olds. In p articular, we sho w b elo w that the p erio d mapping of a doub ly p erio dic closed geo desic, whic h measures the ratio b etw een oscillati on and winding for a geo desics, is algebraic in the parameters of the problem and that it ma y b e explicitly compu ted using the second co v ering asso ciated to the h yp erelliptic curve . In deed, th e 2–dimensional Jac(Γ) is isogenous to the pro du ct of tw o elliptic cur v es E 1 × E 2 . The second co vering p la ys a relev an t role also in the case of elliptic solitons. Airault et al. [4] disco ve r ed a remark able link b etw een the p ole dynamics of the KdV elliptic solutions with the initial data in the form of th e Lam ´ e p oten tial and the dynamics of Calogero–Moser particle s ystem [7]. In the gen us 2 case, the top ological c haracterization of the co vering ramified at P 0 reduces the problem of d escribing the p ole dynamics to th e searc h of solutions of certain algebraic equations related to the co v ering and to the in version of elliptic integral s [5, 33]. Belo w w e first r ecall the d efinition of th e p erio d m ap p ing and some classical results. Then we show how to compute the p erio d mapping explicitly u sing the top ological c haracter of the asso ciated second co vering. Un fortunately there do no exist general theorems whic h c haracterize top ological ly such families of co ve r ings. As an application, we compute the v alue of the p erio d m apping for some sp ecial classes of cov erings in the next section. Closed geo desics on triaxial ellipsoids and the p erio d mapping In the case n = 2 (geod esics on triaxial ellipsoids), Prop osition 2.3 implies that for any fixed c hoice of the semiaxes 0 < a 1 < a 2 < a 3 there is a dense set I ⊂ ] a 1 , a 3 [ \{ a 2 } suc h that for all c ∈ I the hyp er elliptic cur v e Γ : { µ 2 = − λ ( λ − c ) 3 Q i =1 ( λ − a i ) } is Hill. 20 The application c 7→ ϕ ( c ) =    2 H α 2 ω 2 : 2 H α 1 ω 2 , a 1 < c < a 2 < a 3 , 2 H α 1 ω 2 : 2 H α 2 ω 2 , a 1 < a 2 < c < a 3 , (5.1) measures the r atio b et ween osc illation and winding for a geod esics with p arameter c and it is called the p erio d mappin g (see [20]). Comparing th e ab ov e definition with (3.1) and Pr op osition 2.3, it is evident that the geo desic w ith parameter c is closed if and only if ϕ ( c ) is ratio n al. A closed geo desic is cal led simple if it h as no self-in tersections. T o b e simple closed, only a single wind ing is all ow ed; hence ϕ ( c ) m us t b e an in teger greater than one. The follo wing theorems explain un d er w hic h condition there do exist top ologically n on–trivial simp le closed geo desics. Theorem 5.1 [20] L et a 1 < a 2 < a 3 b e fixe d and c ∈ ] a 1 , a 3 [ \{ a 2 } . Then ϕ ( c ) is a monotone de cr e asing function of c . If c ∈ ] a 1 , a 2 [ , then ϕ ( c ) > 1 and lim c → a 2 ϕ ( c ) = 1 . If c ∈ ] a 2 , a 3 [ then ϕ ( c ) < 1 and lim c → a 2 ϕ ( c ) = 1 . Mor e over, let t = a 1 /a 3 b e fixe d and σ = a 2 /a 3 ∈ ] t, 1[ . Then, ϕ ( a 1 ) is a mono- tone incr e asing func tion of σ with upp er limit p a 3 /a 1 and lower limit 1 . Theorem 5.2 [20] On an el lipsoid { 3 P i =1 X 2 i /a i = 1 } , ther e exist non standar d sim- ple close d ge o desics (i.e. simple close d ge o desics differ ent fr om the thr e e princip al el lipses), if and only i f ϕ ( a 1 ) > 2 . Mor e pr e c isely, for e ach i nte ger value ϕ ( c ) ∈ ]1 , ϕ ( a 1 )[ , the pr oje ction of the flow lines yields close d ge o desics which wind onc e ar ound the X 1 –axis while p erforming ϕ ( c ) ma ny oscil lations. Their length is g r e ater than the length of the midd le el lipse in the ( X 1 , X 3 ) –plane. The second co vering In th e case n = 2, Jac(Γ) is isogenous to the pr o duct of tw o elliptic curv es E 1 × E 2 and the second co v ering is ramified at P 0 = (0 , 0) according to the f ollo wing prop osition by Colombo et al Prop osition 5.3 [9] L et Γ b e a genu s 2 cu rve which c overs an e l liptic curve π 1 : Γ 7→ E 1 and let π 2 : Γ 7→ E 2 b e anoth e r c overing so that Jac(Γ) ≈ E 1 × E 2 . Then π i is tangential exactly at the p oints wher e π j is r amifie d i 6 = j . W e briefly tu r n bac k to the double-p erio d icit y condition in the sp ecial case of geod esics on triaxial ellipsoids so to construct directly the second co v erin g asso ciated to the d ouble–p erio d icit y cond ition. Prop osition 5.4 L et Γ = { µ 2 = − λ ( λ − c ) 3 Q i =1 ( λ − a i ) ≡ − λ 4 Q k =1 ( λ − b k ) } b e the genus 2 hyp er el liptic curve asso ciate d to the ge o desics on the triaxial e l lipsoid Q = 21 { X 2 1 /a 1 + X 2 2 /a 2 + X 2 3 /a 3 = 1 } with c austic p ar ameter c . L et α 1 , α 2 , β 1 , β 2 b e the c onventional c anonic al homolo gic al b asis depicte d in Figur e 1 and let ω 1 = dλ/µ , ω 2 = λdλ/µ , b e the b asis of holomorph ic differ entials intr o duc e d in (2.5). Supp ose that on Γ as ab ove, the double p erio dicity c ondition (3.1) and (3.2) holds m 1 I α 1 ω 1 + m 2 I α 2 ω 1 = 2 T , m 1 I α 1 ω 2 + m 2 I α 2 ω 2 = 0 , m ′ 1 I β 1 ω 1 + m ′ 2 I β 2 ω 1 = 2 √ − 1 T ′ , m ′ 1 I β 1 ω 2 + m ′ 2 I β 2 ω 2 = 0 , for some non–zer o r e al T , T ′ . Then, ther e exists a c overing π 2 : Γ 7→ E 2 , r amifie d of or der 3 at P 0 = (0 , 0) and such that π ∗ 2 (Ω 2 ) = κω 2 , wher e Ω 2 is the normalize d holomo rphic differ ential on E 2 and κ is a numeric al c onstant . Pr o of: The double–p erio dicit y conditions (3.1) and (3.2) imply the existence of t wo cycles α = m 1 α 1 + m 2 α 2 and β = m ′ 1 β 1 + m ′ 2 β 2 , suc h that I α ω 2 = 0 , I β ω 2 = 0 . (5.2) In (5.2) it is not restrictiv e to sup p ose that ( m 1 , m 2 ) (r esp ectiv ely ( m ′ 1 , m ′ 2 )), b e relativ e prime in teger n umb ers. Insp ection of (5 .2 ) implies th at all of the p erio ds of ω 2 are in teger multiples of t wo p erio ds S, √ − 1 S ′ of ω 2 and this is su fficien t to p r o ve the existence of a co v ering π 2 : Γ 7→ E 2 . Indeed, let ( m 1 , m 2 ) (resp ective ly ( m ′ 1 , m ′ 2 )), b e relativ e prime in teger n umb ers and let 2 S = H α 1 ω 2 / | m 2 | (resp. 2 √ − 1 S ′ = H β 1 ω 2 / | m ′ 2 | ). By Bezout id en tit y , th ere exist in tegers p j , p ′ j , j = 1 , 2 such that p 1 m 1 − p 2 m 2 = 1 (resp. p ′ 1 m ′ 1 − p ′ 2 m ′ 2 = 1 so that 2 S, 2 √ − 1 S ′ are indeed p erio ds of ω 2 and an y other p erio d H γ ω 2 is an in teger m ultiple of 2 S, 2 √ − 1 S ′ . Let no w fix P 0 = (0 , 0) ∈ Γ as b asep oin t, let z = R P P 0 ω 2 , P ∈ Γ . Then, by P oincar ´ e r educibilit y th eorem, z ∈ T = C / Λ, the one–dimensional torus w ith p erio d lattice Λ generated b y 2 S, 2 √ − 1 S ′ . Finally let P ( z ) ≡ P ( z | S, √ − 1 S ′ ) b e the W eierstrass P -fun ction with half- p erio d s S , √ − 1 S and E 2 :   P ′ ( z )  2 = 4 3 Q k =1  P ( z ) − E k   the elliptic curv e in W eierstrass n ormal form with finite br an ch p oints E 1 = P ( S ), E 2 = P ( S + √ − 1 S ′ ) and E 3 = P ( √ − 1 S ′ ). 22 Then, the cov ering π 2 : Γ 7→ E 2 has degree d and , introdu cing lo cal co ordin ates at P 0 ∈ Γ, it is s tr aigh tforwa r d to verify that it is ramified of order three at P 0 = (0 , 0) (the latter remark implies d ≥ 3).  Of course, by Theorem 3.7, we already know that there exists a co vering π 1 : Γ 7→ E 1 whic h is hyp erelliptic tangen tial at P 0 . T he s econd co v erin g constru cted ab o ve is ramified exac tly at P 0 in agreemen t with Prop osition 5.3. The second cov ering and the p erio d mapping W e now sho w that the top o- logica l t yp e of the second cov ering (whic h is ramified at P 0 of order 3) is natur ally link ed to the top ologica l classification of the associated real closed geod esics (p erio d mapping). Definition 5.5 (top ological c haracterist ic of the second cov ering) The top o- lo gic al char acteristic of a c overing is a se quenc e of four inte ger numb ers ( ν 0 , ν 1 , ν 2 , ν 3 ) which c ount the numb er of Weierstr ass p oints of Γ in the pr ei mage of the four br anch p oint of E 2 , with the exc eption of P 0 = (0 , 0) ∈ Γ , the W e ierstr ass p oint at which the se c ond c overing is r amifie d, and with the usual c onvention that ν 0 is asso ciate d to the br anch p oint of E 2 at infinity. F or a giv en Γ = { µ 2 = − λ 4 Q j =1 ( λ − b j ) } , the compu tation of the p erio d mapping amoun ts to iden tify the t wo in teger n umb ers m 1 , m 2 suc h that m 1 I α 1 ω 2 + m 2 I α 2 ω 2 = 0 . Let π 2 : Γ → E 2 b e th e second co v ering, where E 2 = {W 2 = 4 Z 3 − G 2 Z − G 3 ≡ 4 3 Q i =1 ( Z − E i ) } is represent ed in the canonical W eierstrass form. In our setting th e curv es Γ and E are real with maximal num b er of real connected comp onents, so that it mak es sense to call α the real cycle asso ciated to E . F rom the pro of of Prop osition 5.4, w e kn o w that the p ull-bac k of the h olomorphic differen tial on E 2 is d Z/W = ω 2 . So we ma y conclude that I α i ω 2 = κ i I α d Z /W , i = 1 , 2 , (5.3) where the integ er num b ers κ 1 , κ 2 satisfy m 1 κ 1 + m 2 κ 2 = 0. Finally , κ 1 , κ 2 are uniqu ely asso ciated to the top ologica l c haracteristic of the co v ering π 2 . T o compute them it is sufficient to compute the preimages of the four branc h p oint s E 0 , E 1 , E 2 , E 3 of E 2 . Since in our setting the co vering is real, π − 1 2 ( E i ) are either the branc h p oin ts of Γ or r eal p oin ts on the curve Γ or come in complex conjugate pairs. Then it is self–eviden t that, whenever w e kno w the topological c haracteristic of second co v erin g, we may compute κ 1 and κ 2 . Unfortun ately , we d o not p ossess su c h complete piece of information in th e general case. An ywa y , for an y degree d , there 23 exist a finite num b er of families of hyperelliptic tangen tial co verings so that only a finite num b er of top ological charact er istic are p ossible and, consequently , only a finite num b er of v alues of the p erio d mappin g ma y b e r ealized. In the next section w e discuss the case in whic h th e d egree of the cov ering is either 3 or 4. When the degree of the co vering is 5, there exist t wo families of h y p erelliptic tangenti al co v erings (see [33]) and there exist real d oubly–p erio dic geo d esics asso ciated to suc h co v erings either simple or with 1,2,3 or 4 self–in tersections. Since the complexit y of the computations increases w ith the d egree of the co v ering, we shall rep ort the degree d = 5 case in d etail in a subsequen t p ublication. 6 Examples and applications Explicit examples of hyp erelliptic tangenti al co v ers when the genus of th e h yp er- elliptic curv e is n ≤ 8 hav e b een w orked out (see for instance [38] and references therein). In this section, we imp ose the realit y conditions for algebraic closed geodesics on triaxial ellipsoids for the families of degree d = 3 , 4 hyp erelliptic tangen tial co v ers and we determine the p ossible v alues of the p erio d mapping u sing the top ological c haracter of the second cov ering. F or a comparison w ith the case of elliptic KdV solitons, we r efer to Smirn o v [32, 33] or to Belok olos and Enol’ski [5 ]. Finally in th e last subsection, w e pro v e the existence of doubly–p erio d ic closed geod esics related to degree 2 cov erings with extra automorphisms and we give an explicit example (see Figure 4): in view of Corollary 3.8 in su c h case the cur v e admits also a h y p erelliptic tangen tial co ver, and th en an infin ite num b er of co v er in gs b y a classical theorem by Picard [30]. The same family of co v erin gs has also b een considered by T aimano v [34] in relatio n to elliptic KdV solitons. Remark 6.1 In al l examples, we adopt the fol lowing c onvention: 0 < a 1 < a 2 < a 3 ar e the semiaxes of the triaxial el lipsoid Q = { X 2 1 /a 1 + X 2 2 /a 2 + X 2 3 /a 3 = 1 } and c is the p ar ameter of the c onfo c al quadric to which the ge o desic is tangent, so that the finite br anch p oints of the asso ciate d hyp er el liptic curve Γ ar e { b 0 = 0 < b 1 < b 2 < b 3 < b 4 } = { 0 , c, a i , i = 1 , . . . , 3 } . F or an easier comparison of our r esults with d -elliptic Kd V solitons, we fi r st im- p ose that the hyp erelliptic tangen tial d : 1 co vering ( G , P ∞ ) 7→ ( E , Q ) b e asso ciated to real KdV-solitons, where G : { w 2 = − Q 5 k =1 ( z − z k ) } and P ∞ is the branch p oint of G at infinity . Th en , by Moser–T rub o witz isomorp hism, the curves G and Γ are birationally equ iv alen t and th e follo wing relation among the fi nite b r anc h p oint s z k s of G and th e fin ite branc h p oin ts b j of Γ holds: { z 1 , z 2 , z 3 , z 4 , z 5 } = { β , β + 1 b j , j = 1 , . . . , 4 } , where β = min { z k , k = 1 , . . . , 5 } . (6.1) 24 6.1 Hyp erelliptic tangen t ial co vers of degree 3 . Description of the hyperellipt ic t angen tial co vering and reality problem for doubly–p erio dic closed geo desics The tangen tial 3:1 co v erin g G 7→ E is asso ciated to 3-ell ip tic Kd V solutions and dates bac k to the w orks of Hermite and Halphen ([16]). F or th e closed geo desics problem, w e r equire that th e gen u s 2 curve Γ is birationally equiv alent to G =  w 2 = − 1 4 (4 z 3 − 9 g 2 z − 27 g 3 )( z 2 − 3 g 2 )  whic h co v ers the elliptic curv e E 1 = { W 2 = 4 Z 3 − g 2 Z − g 3 } , where the co vering is giv en b y the relations Z = − 1 9 z 3 − 27 g 3 z 2 − 3 g 2 , W = 2 27 w ( z 3 − 9 g 2 z + 54 g 3 ) ( z 2 − 3 g 2 ) 2 . The holomorphic differenti al on E 1 is the pull–bac k of the holomorphic differentia l d Z W = − 3 2 z dz w on G . The z j s are r elated to the branch p oin ts of E 1 , e j , j = 1 , . . . , 3 , by β ≡ z 1 = − p 3 g 2 < z 2 = 3 e 1 < z 3 = 3 e 2 < z 4 = 3 e 3 < z 5 = p 3 g 2 , where e 1 < e 2 < e 3 . Prop osition 6.2 L et 0 < a 1 < a 2 < a 3 and c ∈ ] a 1 , a 3 [ \{ a 2 } b e given. Then the ge o desic flow on the el lipsoid Q tangent to the c onfo c al quadric Q c is doubly p erio dic and r elate d, up to bir ational tr ansformat i on, to the 3:1 c overing G → E 1 if and only if 1 c 2 + 1 a 2 2 + 1 a 2 3 − 2  1 ca 2 + 1 ca 3 + 1 a 2 a 3  = 0 , a 1 = 3 ca 2 a 3 2( a 2 a 3 + c ( a 2 + a 3 )) . (6.2) If (6.2) holds, then the b r anch p oints on E 1 ar e { e 1 , e 2 , e 3 } =  2 a 1 − c 6 a 1 c , 2 a 1 − a i 6 a 1 a i , i = 2 , 3  , β = − 1 2 a 1 and g 2 = − 1 12 a 2 1 . (6.2) may b e inv erted and we get a 1 , a 3 parametrically in fun ction of a 2 , c or a 2 , c in fun ction of a 1 , a 3 . Corollary 6.3 L et a 2 , c > 0 b e given and a 2 6 = c . Then (6.2) is e qu i v alent to a 3 =  1 √ a 2 − 1 √ c  − 2 , a 1 = 3 a 2 c 4( a 2 + c − √ ca 2 ) . L et 0 < a 1 < a 3 b e given. Then (6.2) is e quivalent to 1 a 2 , 1 c = ± 1 2 √ a 3 + r 4 3 a 1 − 3 4 a 3 . 25 The second co v ering and t he p e rio d mapping Let Γ = { µ 2 = − λ 4 Q k =1 ( λ − b k ) } , then the second 3:1 co v erin g π 2 : Γ → E 2 has top ological charact eristic (0 , 3 , 1 , 1) (see [5, 33 ]). In this case the explicit expression of the co vering Π 2 : G 7→ E 2 is known [33] and it is giv en by the maps Z = − 1 4 (4 z 3 − 9 g 2 z − 9 g 3 ) , W = − 1 2 w  4 z 2 − 3 g 2  and the mo duli of E 2 are G 2 = 27 4 ( g 3 2 + 9 g 2 3 ), G 3 = − 243 8 g 3 (3 g 2 3 − g 3 2 ). Th e fi nite branc h p oin ts of E 2 are E 1 = − 9 / 2 g 2 , E 2 = 9 / 4 g 3 + 3 / 4 g 2 √ 3 g 2 and E 3 = − E 1 − E 2 and satisfy E 2 < E 1 < E 3 . Using the birational transformation z = 1 /λ − √ 3 g 2 , we find the explicit expr es- sion of the co vering π 2 : Γ 7→ E 2 . It is ramified of order 3 at b 0 = 0 (and m ap p ed to infinity by π 2 ), that is π − 1 2 ( E ∞ ) = { P 0 , P 0 , P 0 } and π − 1 2 ( E 1 ) = { b 2 , b 3 , b 4 } , π − 1 2 ( E 2 ) = { b 1 , P ± } , π − 1 2 ( E 3 ) = { b ∞ , Q ± } , where b ∞ denotes the infinite ramification p oin t of Γ, b j s are the fi nite r amification p oints of Γ (with a sligh t abu se of n otation, w e u s e the s ame symb ol for the p oint on the curve and its λ co ordinate), P ± and Q ± are the real p oints on Γ suc h that λ ( P ± ) = 2 / √ 3 g 2 and λ ( Q ± ) = 2 / √ 27 g 2 . Finally , it is ea s y to c hec k that λ ( P ± ) ∈ ] b 3 , b 4 [ , λ ( Q ± ) ∈ ] b 2 , b 3 [ , so that I α 1 ω 2 I α 2 ω 2 = Z b 2 b 1 ω 2 Z b 4 b 3 ω 2 = 2 Z E 1 E 2 d Z / W 4 Z E 1 E 2 d Z / W = I α d Z / W 2 I α d Z / W = 1 2 , and fi nally , comparing the defin ition of p erio d mapping (5.1 ) with (5.3), w e conclude that the p erio d mapping is eit h er 2 : 1 or 1 : 2. Also the dual curve Γ ′ defined in T heorem 3.4 is a h yp erelliptic tangenti al cov er of degree d = 3 and the branch p oin ts of Γ ′ still satisfy Prop osition 6.2, so that the algebraic r eal closed geo desics asso ciated to the d ual cur v e ha ve p erio d m apping 2 : 1 or 1 : 2. W e ha ve thus prov en the follo w ing Lemma 6.4 The close d ge o desics asso ciate d to a r e al curve Γ which is a 3:1 hyp er- el liptic al tangential c over, either have p erio d mapping 2 : 1 or 1 : 2 . 26 6.2 Hyp erelliptic tangen t ial co vers of degree 4 . Description of the hyperellipt ic t angen tial co vering and reality problem for doubly–p erio dic closed geo desics In this case, we require Γ to b e bira- tionally equiv alen t to G = n w 2 = − Q 5 i =1 ( z − z i ) o , wh er e z 1 = 6 e j , z 2 , 3 = − e k − 2 e j ± 2 q ((7 e j + 2 e k )( e j − e k ) , z 4 , 5 = − e l − 2 e j ± 2 q ((7 e j + 2 e l )( e j − e l ) . (6.3) G co vers the elliptic curve E 1 with mo duli g 2 , g 3 , E 1 = { W 2 = 4 Z 3 − g 2 Z − g 3 = 4 Q 3 s =1 ( Z − e s ) } ⊂ ( Z, W ) , and the co v ering is give n b y the relations Z = e j + ( z 2 − 3 z e j − 72 e 2 j − 27 e l 2 e k ) 2 4( z − 6 e j )(2 z − 15 e j ) 2 , d Z W = (2 z − 3 e j ) dz w . The explicit expr ession of this co ve r ing has b een foun d b y Belok olos and Enolski [5] (see also [33]). The realit y cond ition for z i , i = 1 , . . . , 5 is H 2 j = 3 Q k 6 = j ( e j − e k ) > 0, that is e j is either e 1 or e 3 in (6.3). If e j = e 1 in (6.3), then β = 6 e 1 ; if e j = e 3 in (6.3), th en β = − e 2 − 2 e 3 − 2 p (7 e 2 + 2 e 3 )( e 3 − e 2 ). In b oth cases, we give n ecessary and sufficient conditions using the follo wing n otation f 1 = 1 /b 1 , f 2 = 1 /b 2 , f 3 = 1 /b 3 , f 4 = 1 /b 4 . –1 –0.5 0 0.5 1 x(t) –1 –0.5 0 0.5 1 y(t) –10 –8 –6 –4 –2 0 2 4 6 8 10 z(t) –1 –0.5 0 0.5 1 x(t) –2 –1 0 1 2 y(t) –10 –8 –6 –4 –2 0 2 4 6 8 10 z(t) Figure 2. Prop osition 6.5 L et 0 < f 4 < f 3 < f 2 < f 1 as define d ab ove. Then the ge o desic flow on the el lipsoid Q tangent to the c onfo c al quadric Q c is doubly p erio dic and 27 r e late d, up to bir ational tr ansformat i on, to the 4:1 c overing G → E 1 intr o duc e d ab ove, i f and only if either c onditio n (A) or (B) b elow is fulfil le d: ( A )    81( f 2 2 + f 2 3 ) + 62 f 2 2 f 2 3 + 36( f 1 + f 4 ) 2 − 108( f 2 + f 3 )( f 1 + f 4 ) = 0 , 81( f 2 1 + f 2 4 ) + 62 f 2 1 f 2 4 + 36( f 2 + f 3 ) 2 − 108( f 2 + f 3 )( f 1 + f 4 ) = 0 . ( B )    100 f 4 f 3 + 8 f 1 ( f 4 + f 3 + 8 f 1 − 9 f 2 ) − (9 f 4 + 9 f 3 − 6 f 2 ) 2 = 0 , (6 f 4 + 6 f 3 − 9 f 2 ) 2 + 8 f 1 (9 f 4 + 9 f 3 − 8 f 1 − f 2 ) = 0 If (A) hold s, then β = 6 e 1 and the br anch p oints of E 1 ar e e 1 = − 1 30 ( f 1 + f 2 + f 3 + f 4 ) , e 2 = 4 15 ( f 1 + f 4 ) − 7 30 ( f 2 + f 3 ) , e 3 = − e 2 − e 1 . –0.3 –0.2 –0.1 0 0.1 0.2 0.3 x(t) –0.3 –0.2 –0.1 0 0.1 0.2 0.3 y(t) –2 –1 0 1 2 z(t) –0.3 –0.2 –0.1 0 0.1 0.2 0.3 x(t) –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 y(t) –2 –1 0 1 2 z(t) Figure 3. If (B) hol ds, then the br anch p oints of E 1 ar e e 3 = 2 15 f 1 − 1 30 ( f 2 + f 3 + f 4 ) , e 2 = − f 1 15 + 4 f 2 15 − 7 30 ( f 3 + f 4 ) , e 1 = − e 2 − e 3 and β = − 1 5 ( f 1 + f 2 + f 3 + f 4 ) . Conditions (A) and (B) ma y b e expressed in the follo wing equiv alent w a y Prop osition 6.6 ( A ′ ) L et σ ± > 0 and such that 2 / 3 σ + < σ − < σ + . L et γ + = 9 25  3 2 σ + − σ −  2 , γ − = 9 25  3 2 σ − − σ +  2 . 28 Then 0 < f 4 < f 3 < f 2 < f 1 satisfy (A) i f and only i f f 2 , f 3 (r esp. f 1 , f 4 ) ar e the r o ots of x 2 − σ + x + γ + = 0 , (resp . x 2 − σ − x + γ − = 0 . In this c ase, the mo duli of the el liptic curve ar e g 2 = 19 75 ( σ + + σ − ) 2 − σ + σ − , g 3 = 28 3375 ( σ + + σ − ) 3 − σ + σ − 30 ( σ + + σ − ) . ( B ′ ) L et − σ + < σ − < 0 and define γ + = 1 5 q 16 σ 2 + − 18 σ − σ + − 9 σ 2 − , γ − = 1 5 q 16 σ 2 + + 2 σ + σ − − 14 σ 2 − . Then 0 < f 4 < f 3 < f 2 < f 1 satisfy (B) if and only if f 1 = σ + + γ + , f 2 = 2 γ + , f 3 = σ − 2 + γ + + γ − , f 4 = σ − 2 + γ + − γ − . If the ab ove e quation is satisfie d, the mo duli of the el liptic curve ar e g 2 = 19 75 σ 2 − − 2 75 σ − σ + + 4 75 σ 2 + , g 3 = 28 3375 σ 3 − + 8 3375 σ 3 + − 37 1125 σ 2 − σ + − 2 1125 σ − σ 2 + . Prop osition 6.7 L et Γ = { µ 2 = − λ 4 Q k =1 ( λ − b k ) } b e a r e al 4:1 hyp er el liptic tangen- tial c over ve rifying Pr op osition 6.5 (A). Then the br anch p oints of the dual curve Γ ′ intr o duc e d in The or em 3.4 satisfy Pr op osition 6.5 (B) and vic eversa. The second co v ering and t he p e rio d mapping Let Γ = { µ 2 = − λ 4 Q k =1 ( λ − b k ) } , then the second 4:1 co v erin g π 2 : Γ → E 2 has top ological charact eristic (1 , 2 , 2 , 0) and its explicit exp r ession is gi ven in [5, 33]. Let E 2 = {W 2 = 4 3 Q i =1 ( Z − E i ) } , E 1 < E 2 < E 3 , then pro ceeding as for the case of the degree 3:1 cov ering, after some ugly and straigh tforward computations, we arriv e to the follo win g conclusion. If Prop osition 6.5 (A) holds, b 0 = 0 is a order 3 ramification p oint map p ed to infinity by π 2 , b 1 , b 4 ∈ π − 1 2 ( E 1 ), b 2 , b 3 ∈ π − 1 2 ( E 2 ) and the infinit y p oint of Γ maps to the infi nit y of E 2 . Finally , computing the solutions to the equatio n ( λ, µ ) = π − 1 2 ( E j ), j = 1 , 2 w e fin d real p oin ts with λ coordinate in ] b 3 , b 4 [, and we conclude that I α 1 ω 2 I α 2 ω 2 = 2 Z b 2 b 1 ω 2 2 Z b 4 b 3 ω 2 = 2 Z E 1 E 2 d Z / W 6 Z E 1 E 2 d Z / W = I α d Z / W 3 I α d Z / W = 1 3 . That is, th e p erio d mapp in g is either 3 : 1 or 1 : 3. 29 If Prop osition 6.5 (B) h olds, using Prop osition 6.7 and pro ceeding as ab o v e, w e get I α 1 ω 2 I α 2 ω 2 = 2 Z b 2 b 1 ω 2 2 Z b 4 b 3 ω 2 = 2 Z E 1 E 2 d Z / W 4 Z E 1 E 2 d Z / W = I α d Z / W 2 I α d Z / W = 1 2 . and we conclude th at the p erio d mapping is either 2 : 1 or 1 : 2. W e hav e th u s pro ven Prop osition 6.8 The close d ge o desics asso ciate d to a curve Γ which is a 4:1 hy- p er el liptic al tangential c over, have p erio d mapping 3 : 1 or 1 : 3 in c ase Pr op osition 6.5 (A ) holds, and have p erio d mapping 2 : 1 or 1 : 2 in c ase Pr op osition 6.5 (B) holds. Figures 2 and 3: In figure 2 we p resen t closed geo desics with p erio d mapping 1 : 3 ( a 1 < a 2 < c < a 3 ) and 3 : 1 ( a 1 < c < a 2 < a 3 ) asso ciated to the h y p erelliptic curv e Γ = { µ 2 = − λ ( λ − 1 . 453)( λ − 1 . 483)( λ − 4 . 434)( λ − 84 . 967) } , whic h is a 4:1 hyp er elliptic tangen tial co v er corresp onding to σ + = 2 . 7, σ − = 2 . 1 so that Prop osition 6.5 (A) is satisfied. In figure 3 we present closed geo desics with p er io d mapp ing 1 : 2 ( a 1 < a 2 < c < a 3 ) and 2 : 1 ( a 1 < c < a 2 < a 3 ) asso ciated to the h y p erelliptic cur v e Γ = { µ 2 = − λ ( λ − 0 . 099 6)( λ − 0 . 1012)( λ − 0 . 150)( λ − 4 . 5510) } , whic h is a 4:1 hyp er elliptic tangen tial co v er corresp onding to σ + = − 3, σ − = 5 . 1 so that Prop osition 6.5 (B) is satisfied. 6.3 Doubly–p erio dic closed geo desics related t o degree 2 co v erings with extra automorphisms In this section w e pr o ve the existence of a family of doubly–p erio d ic closed geod esics on triaxial ellipsoids parametrized by τ 2 ∈ Q related to the family of gen us tw o h yp erelliptic cu rv es Γ which co v ers 2:1 tw o isomorphic elliptic curve s E 1 , 2 (this family of co v erings has also b een considered in relation to d oubly–p erio d ic KdV solutions b y I. T aimano v [34 ]). The parameter τ is the mo duli of the elliptic cur ve E 1 . Sin ce it is not p ossible to determine algebraically the br anc h p oin ts of an elliptic cur v e in function of the mo duli or v iceve r sa, the condition on τ is transcendental. Ho wev er Theorem 3.7 implies that for su c h v alues of the p arameter τ 2 , Γ is also a hyp erelliptic tangen tial co v er of another curv e E 3 , so that in p rinciple it sh ould b e p ossib le to express such condition also algebraically . Indeed w e hav e b een able to w ork out an exp licit example (Figure 4) associated to the real in tersection of this one parameter family of d egree 2 co verings with extra automorphisms with th e t wo-paramete r family of degree 3 hyp erelliptic tangen tial co ve r s c haracterized in subsection 4.1. 30 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 x(t) –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 y(t) –3 –2 –1 0 1 2 3 z(t) Figure 4. Description of t he co vering T h e hyperelliptic cu rv e G α =  w 2 = z ( z 2 − α 2 )( z 2 − 1 /α 2 )  , co v ers 2:1 the elliptic curv e E 1 =  W 2 = Z ( Z − 1)( κ 2 α Z − 1) }  , κ 2 α = ( α + 1) 2 2( α 2 + 1) , and the co vering Π 1 : G α 7→ E 1 is giv en b y Z = − 2(1 + α 2 ) z α ( z − α )( z − 1 /α ) , W = r − 2(1 + α 2 ) α ( z + 1) w ( z − α ) 2 ( z − 1 /α ) 2 , and, moreo ver, d Z W = − q − 2(1+ α 2 ) α ( z − 1) dz w . There exists a second 2:1 co ver Π 2 : G α 7→ E 2 , with E 2 = n ˜ W 2 = ˜ A α ˜ Z ( ˜ Z − 1)( κ 2 α ˜ Z − 1) o , ˜ A α = − 2( α + 1) 2 ( α 2 + 1) ( α − 1) 4 , ˜ Z = ( z − α )( z − 1 /α ) ( z − 1) 2 , ˜ W = r − 2 α 2 + 1 α y ( x − 1) 3 and d ˜ Z ˜ W = ( α − 1) 2 √ 2 α ( α 2 +1) ( z +1) dz w . Clearly E 1 and E 2 are isomorphic sin ce they ha ve the same j –inv arian t (see f or instance [2]). No w, let α > 1. Under the birational transformation, λ = 1 / ( z + α ) , µ = y √ 1 − α 4 ( z + α ) 3 , 31 G α is equiv alen t to Γ α =  µ 2 = − λ ( λ − 1 2 α )( λ − 1 α )( λ − α α 2 − 1 )( λ − α α 2 + 1 )  . (6.4) Since 0 < 1 2 α < α α 2 + 1 < 1 α < α α 2 − 1 , Γ α ma y b e inte r preted as the h yp er- elliptic curve asso ciated either to th e geo desics on the ellipsoid Q 0 of semiaxes 1 2 α , α α 2 + 1 , α α 2 − 1 and tangen t to th e confo cal qu ad r ic Q c , c = 1 α , or to the geod esics on the ellipsoid Q 0 of semiaxes 1 2 α , 1 α , α α 2 − 1 and tangent to the con- fo cal quadr ic Q c , with c = α α 2 + 1 . The family of h yp erelliptic curv es Γ α is rather exceptional. Ind eed, the birational transformation ρ = a 1 λ/ ( λ − a 1 ) introd uced in Lemma 3.3, just p erm u tes the branch p oints so that Γ coincides with its dual curve Γ ′ . Using Theorem 3.4, w e immediately get Prop osition 6.9 The r e al ge o desics asso ciate d to Γ α ar e close d if and only if they ar e doubly–p erio dic. In the latter c ase Γ α c oincides with its dual. A t ranscenden tal condition for doubly–p erio dic closed geo desics W e now discuss the existence of su c h d oubly–p erio d ic closed geo desics for Γ α . Usin g the ab o ve formulas it is easy to c hec k that d Z q Z ( Z − 1)( Z − κ − 2 α ) = 2 ρ α (( α + 1) λ − 1) dλ µ , d ˜ Z q ˜ Z ( ˜ Z − 1)( ˜ Z − κ − 2 α ) = 2 iρ α (( α − 1) λ − 1) dλ µ , where ρ α = q (1+ α 2 ) 4 α ( α − 1)( α 2 +1) . No w let P 1 = ( λ 1 , µ 1 ) , P 2 = ( λ 2 , µ 2 ) ∈ Γ a and set U i = 2 Z Π 1 ( P i ) ∞ d Z q Z ( Z − 1)( Z − κ − 2 α ) , ˜ U i = 2 Z Π 2 ( P i ) ∞ d ˜ Z q ˜ Z ( ˜ Z − 1)( ˜ Z − κ − 2 α ) , ( i = 1 , 2) . Then, the qu adrature of the geodesics flo w 2 X i =1 Z P i P 0 dλ µ = s + const., 2 X i =1 Z P i P 0 λdλ µ = const., is equiv alen t to U 1 + U 2 = − ρ α s + c, ˜ U 1 + ˜ U 2 = − √ − 1( ρ α s + ˜ c ) , 32 with c, ˜ c constan ts. Using th e addition theorem for elliptic integrals, the ab ov e equations ma y b e inv erted and w e get P ( U 1 | τ α )+ P ( U 2 | τ α ) = P ( − ρ α s + c | τ α ) , P ( ˜ U 1 | τ α )+ P ( ˜ U 2 | τ α ) = P ( − √ − 1( ρ α s + ˜ c ) | τ α ) , (6.5) where 0 < τ α < 1 is th e m o duli of E 1 . Using the ident ity P ( √ − 1 U | τ α ) = P  U | − 1 τ α  , (6.5) is equiv alen t to P ( U 1 | τ α ) + P ( U 2 | τ α ) = P ( − ρs + c | τ α ) , P ( ˜ U 1 | τ α ) + P ( ˜ U 2 | τ α ) = P ( − ρs + ˜ c | − 1 τ α ) . (6.6) Then, the geodesic is doubly p erio dic if and only if τ 2 α ∈ Q and , in suc h a case, the parameter s may b e eliminated from (6.6) u sing th e addition theorem for elliptic functions. In view of theorem 3.7, then Γ α also p ossesses a h yp erelliptic tangenti al co v er of conv enien t degree d (actually it p ossesses an infin ite num b er of co v er in gs follo wing [30]). W e ha v e th us p ro ven the follo w in g Theorem 6.10 L et Γ α b e the one p ar ameter family of hyp er el liptic curves describ e d ab ove and let 0 < τ α < 1 b e the mo duli of E 1 . Then, the ge o desics asso ciate d to Γ α ar e doubly–p erio dic i f and only if τ 2 α ∈ Q . In the latter c ase, ther e exist an i nte ger d ≥ 3 and an e l liptic curve E ( d ) such that (Γ α , P 0 ) is also d : 1 hyp er el liptic tangential c over over E ( d ) . F rom Theorem 6.10 and the argum en t used to pro ve Pr op osition 6.9, we imme- diately conclude th e follo wing. Corollary 6.11 Supp ose that the g e o desics asso ciate d to Γ α ar e doubly–p erio dic. Then the p erio d map ping of the r e al and i maginary close d ge o desics ar e either e qual or r e cipr o c al to e ach other. The ab ov e Corollary settles quite restrictiv e conditions on the p ossible hyperel- liptic tangen tial co verings asso ciated to Γ α . F or instance there cannot exist d = 4 h yp erelliptic tangen tial co verings asso ciated to Γ α , since the curve and its dual w ould p ossess d ifferen t v alues of the p erio d mappin g for that degree of the co vering (compare Prop ositions 6.7 and 6.8 for the d = 4 hyperelliptic co verings with Prop o- sition 6.9). Belo w we construct explicitly a d = 3 hyper elliptic tangen tial co vering in the family Γ α . Figure 4. W e sh ow an example of doubly–p erio dic closed geo d esic asso ciated to a co vering satisfying Theorem 6.10. T his example p ossesses rather exce p tional and in triguing pr op erties. Let α = q 2 / √ 3 and Γ α as in (6.4), then Γ α co v ers 2:1 the elliptic curve E (2) 1 =  W 2 = Z ( Z − 1)( κ 2 α Z − 1) , }  , where κ 2 α = 1 / 2 + 2 q 2 √ 3 − q 6 √ 3 . 33 Moreo v er, the br anc h p oints of Γ α , a 1 = 1 / (2 α ) , a 2 = α/ ( α 2 + 1) , a 3 = α/ ( α 2 − 1) , c = 1 /α, also satisfy (6.2) in Prop osition 6.2, that is Γ α is a degree 3 h yp erelliptic tangen tial co v er o ver the elliptic curv e E (3) 1 = { w 2 = 4 z 3 − 2 / 9 √ 3 z } . W e remark that the j -in v ariant of E (3) 1 tak es the exceptional v alue 1728, that is the elliptic curve E (3) 1 p ossesses non trivial automorphisms of order t wo (see [2] and references therein). Finally for the second 3:1 co ver w e find G 2 = g 2 , G 3 = g 3 = 0, that is E (3) 2 = E (3) 1 ! As exp ected the closed geo d esics hav e p erio d mapping 1:2 (one self–int er s ection), if w e exc hange c and a 2 w e get p erio d mapp ing 2:1 and s imple closed geodesics. Ac kno wle dgements I w armly thank A. T r eibich for many clarifying discussions concerning the top ologica l classification of h yp erelliptic tangent ial co v erings. I also w armly thank Y u. F edoro v, E. Previato and P . v an Mo erb ek e f or many helpfu l discussions d uring the prep aration of the pap er. Finally , I thank F. Calogero, B. Dubro vin , V. Enols’kii, F. Gesztesy , V. Matv eev, A. S mirno v and I. T aimano v for their int erest in this r esearc h. This w ork h as b een p artially supp orted by the E urop ean Science F oundation Programme MISGAM (Metho d of I n tegrable Systems, Geometry an d Applied Math- ematics) the R TN ENIGMA and PRIN2006 ”Meto di geometrici nella teoria delle onde non lineari ed applicazioni”. Th e fi gu r es and the computations in the example section hav e b een carried out with the help of Ma p le pr ogram. References [1] Ab enda S., F edoro v, Y u . N. Closed geo desics and b illiards on quadrics r elated to elliptic K dV solutions. L ett. Math. Phys. 76 (2006), no. 2-3, 111–134. [2] Accola R.D.M., Previato E. Cov ers of tori: gen us 2. L ett. Math. Phys. 76 (2006), no. 2-3, 135–161. [3] Aud in M. Cour b es alg ´ ebriques et syst ` emes int ´ egrables: g ´ eo d´ esiques des quadriques. Exp o. Math. 12 , n o.3 (1994), 193–226 [4] Airault H., McKean H. P ., Moser J. Rational and elliptic solutions of the KdV equation and a related many-bo d y pr oblem. Commun. Pur e Appl. Math. 30 , (1977 ), 94–148 [5] Belok olos E.D., Enol’ski V.Z. Isosp ectral deformations of elliptic p otenti als. R u ssian Math. Surveys 44 (19 89), no. 5, 191 –193. 34 [6] Belok olos E.D., Bob enko A.I., En ol’ski V.Z., Its A.R., and Matve ev V.B. Algebr o-Ge ometric Appr o ach to Nonline ar Inte gr able Equations. Spr inger S e- ries in Nonlinear Dynamics. Springer–V erlag 1994. [7] Calogero F. Exactly solv able one-dimensional many-bo dy pr oblems. L ett. Nuovo Cimento 13 (1975), 411 –415 [8] Ch asles M. Les lignes g ´ eo d´ esiqu es et les lignes d e courbure des sur faces du second degre ´ e. Journ. de Math. 11 , 5-20 (1846). [9] Colombo E., Pirola G.P . and Previato E. Densit y of elliptic solitons. J. r eine angew. Math. 451 (1994), 16 1–169. [10] Dubr o vin B.A. Periodic p roblems for the Korteweg de V ries equation in the class of fi nite–band p otent ials. F unct. Anal. Appl. 9 (197 6), 215-223 . [11] Dubr o vin B. A., No vik ov, S. P . A p erio d ic problem f or the Kortew eg-de V r ies and Sturm -Liouville equations. Their connection w ith algebraic geometry . (Rus- sian) Dokl. Akad. Nauk SSSR , 219 (19 74), 531—534 . [12] F ark as H.M., K ra I. Rieman n surfac es , Graduate T exts in Mathematics 71 , Springer–V erlag (1980). [13] F edoro v, Y u. N. Algebraic closed geo desics on a triaxial ellipsoid. R e gul. Chaotic Dyn. 10 (2005), no. 4, 463 –485. [14] Gesztesy F., W eik ard R. Elliptic algebro-geometric solutions of th e KdV and AKNS hierarchies - an analytic approac h. Bul l. Amer. Math. So c. 35 (1998), 271-3 17. [15] Gunn ing R. it L e ctur es on Riemann Surfac es. Jac obi varieties. Prin ceton Uni- v ersity Press. (1972) [16] Hermite C. Oeuvr es de Charles Hermite. V ol. III , Gauthier–Villars, P aris, 1912. [17] Igusa J.-I. Arithmetic v ariet y of mo duli for gen us tw o. Ann. Math. 72 , 612–6 49 (1960 ). [18] Its A., Matv eev V. The p erio d ic Kortew eg de V ries equation. F unkt. Anal. Pril. 9 , (1975), 67–70. [19] Jacobi C . V orlesungen ¨ u b er Dynamik, Suppl e mentb and. Berlin (188 4). [20] Klingenberg W. R i emannian ge ometry. de Gruyter Stud ies in Mathematics 1 (1982 ). [21] Kn¨ orrer H. Geo desics on the ellipsoid. Invent. Math. 59 (1980), 119 –143 [22] Kn¨ orrer H. Geod esics on quad r ics and a mec hanical problem of C. Neumann. J. R eine Angew. Math. 334 (1982 ), 69–78. 35 [23] Krichev er I. Elliptic solutions of the Kadom tsev–Pe tviashvili equ ation and in te- grable systems on n particles on the line. F unc t. Ana l. Appl. 14 (1980), 45–54. [24] Lax P . Pe r io dic solutions of the K ortew eg de V r ies equation. Commun. Pu r e Appl. Mathem. 28 (1975), 14 1–188. [25] Mc Kean H.P ., V an Mo erb erke P . The sp ectrum of the Hill equation. Inv. Math. 30 , (1975), 217–274. [26] Mc K ean H.P ., V an Moerb erke P . Hill and T o d a curves. Commun. Pur e Appl. Mathem. 33 , (1980), 23– 42. [27] Moser J. V arious Asp ects of I ntegrable Hamiltonian S y s tems. In: C.I.M.E. L e ctur es, Br essanone, Italy , (1978) [28] Moser J. Geometry of quadr ics and sp ectral theory . The Chern Symp osium 1979 (Pro c. Internat. Symp os., Berk eley , Calif., 1979), pp. 147–1 88, Spr inger, New Y ork-Berlin, (1980) . [29] No viko v S.P . The p erio dic problem for th e Kortew eg–de V r ies equation. F unkt. Ana l. Pril. 8 (1974), 54– 66. [30] Picard E. Sur la r´ edu ction d u nom br e del p ´ erio des des int ´ egrales ab ´ eliennes et, en particulier, dans le cas de courb es du second genre. Bul l. So c. Math. F r. 11 (1883 ), 25–53. [31] Po in car ´ e H. Su r la r ´ eduction d es int ´ egrale s ab´ eliennes. Bul l. So c. Math. F r. 12 (1884 ), 124–143. [32] Smir n o v A. O. Elliptic solutions of the KdV equation. Math. Z. 45 , (1989), 66–73 [33] Smir n o v A. O. Fin ite-gap elliptic s olutions of the KdV equation. A cta Appl. Math. 36 , (1994), 12 5–166 [34] T aimano v I. A. On the tw o-gap elliptic p oten tials. A cta Appl. Math. 36 (1994), no. 1-2, 119–124. [35] T reibic h A., V erdier J.-L. T angen tial cov ers and sums of 4 triangular num b er s . C. R. A c ad. Sci. Paris 311 , (1990), 51–5 4 [36] T reibic h A., V erdier J.-L. Solitons elliptiques. With an app end ix by J. Oesterl. Pr o gr. Math. , 88 , The Grothendiec k F estsc hr ift, V ol. I I I, 437–48 0, Birkhuser Boston, Boston, MA, (199 0). [37] T reibic h A, V erdier J .-L. Revˆ etement s exceptionnels et somme de quatre nom- bres triangulairs. Duke Math. J. 68 (199 2), 217–23 6. [38] T reibic h A. Hyp erelliptic tangentia l co v ers, and finite-gap p oten tials. (Russian) Usp ekhi Mat. Nauk 56 (2001) , no. 6(342), 89–136; translation in Russian Math. Surveys 56 (2001), no. 6, 1107–11 51 36 [39] V erdier J.-L. New elliptic solitons. Algebr aic ana lysis, V ol. II , Academic Pr ess, Boston, MA, (19 88), 901–9 10. [40] W eierstrass K. ¨ Ub er die geo d¨ atisc hen Linien auf dem dr eiac hsigen Ellipsoid. In : Mathematische Werke I , 257–26 6 37