Riemannian metrics having common geodesics with Berwald metrics

Riemannian metrics ha ving common geo desics with Berw ald metrics Vladimir S. Matv eev ∗ Abstract In Theorem 1, we generalize some results of Szab´ o [Sz1, Sz2] for Berwald metrics that are not necessarily strictly convex: we show t hat for every Berwald metric F there alwa ys exists a Riemannian metric affine equiv alent to F . As an application w e show (Coro llary 3) that every Ber wald pro jectively flat metric is a Minko wsk i metric; this sta tement is a “Berwald” version of Hilber t’s 4th proble m. F urther, w e inv estiga te geo desic equiv alence of B e rw a ld metrics. Theorem 2 gives a system of PDE that has a (nontrivial) solution if and only if the given essentially Berwald metric admits a Riemannian metric that is (nontrivially) geo desically equiv alent to it. The system of PDE is linea r a nd o f Ca uc hy-F rob enius type, i.e., the deriv atives of unkno wn functions ar e explicit expressions of the unknown functions. As an applica tion (Cor o llary 2), w e obtain that g eodesic equiv a le nc e of an essentially Berwald metric and a Riemannian metric is alwa y s affine equiv alence provided b oth metrics ar e complete. 1 Definitions and results A Finsler metr ic on a sm ooth manifold M is a fu nction F : T M → R ≥ 0 suc h that: 1. It is smo oth on T M \ T M 0 , w here T M 0 denotes the zero section of T M . 2. F or ev ery x ∈ M , the restrictio n F | T x M is a n orm on T x M , i.e., for ev ery ξ , η ∈ T x M and for ev ery nonnegativ e λ ∈ R we h a v e (a) F ( λ · ξ ) = λ · F ( ξ ), (b) F ( ξ + η ) ≤ F ( ξ ) + F ( η ), (c) F ( ξ ) = 0 = ⇒ ξ = 0. W e alw a ys assume that n := dim( M ) ≥ 2. W e d o not require that (the restriction of ) the function F is strictly con v ex. In th is p oint our defin ition is more general th an the usu al definition. In addition we do not assu me that the metric is reversible, i.e., we d o not assume that F ( − ξ ) = F ( ξ ) . Some s ta ndard references for Finsler geometry are [Al2, BCS, BBI, S h1 ]. ∗ Institute of Mathematics, FSU Jena, 07737 Jena German y , matveev@minet.uni-jena.de 1 Example 1 (Riemannian metric) . F or every Riemannian metric g on M , the function F ( x, ξ ) := q g ( x ) ( ξ , ξ ) is a Finsler metric. A Finsler metric is Berwald , if there exists a symmetric affine connection Γ such that th e parallel transp ort w ith resp ect to this connection preserv es the fun ct ion F . In this case, w e call the connection Γ the asso ciate d c onne ction . Riemannian m et rics are alw ays Berwald. F or them, the asso ciated connection coincides w ith the Levi-Civita connection. W e say that a Finsler metric is essential ly Berwald , if it is Berw ald, but not Riemannian. The simplest examples of essentia lly Berw ald metrics are Mink o wski metrics. Example 2 (Mink o ws ki metric) . Consider a smo oth norm on R n , i.e., a smo oth function p : R n → R ≥ 0 satisfying 2a, 2b, 2c . We c anonic al ly identify T R n with R n × R n with c o or dinates ( x 1 , ..., x n | {z } x ∈ R n , ξ 1 , ..., ξ n | {z } ξ ∈ T x R n ) . Then, F ( x, ξ ) := p ( ξ ) i s a Finsler metric. We se e that the metric is tr anslation invariant. Henc e, the standar d flat c onne ction pr eserves it, i.e., it i s a Berwald metric. If the norm p do es not satisfy the p ar al lelo gr am e quality, the M i nkow ski metric is essential ly Berwald. Let F 1 , F 2 b e Finsler metrics on the same manifold. W e say that F 1 is ge o desic al ly e quivalent (or pr oje ctively e q u ivalent ) to F 2 , if ev ery F 1 − geo d esic, considered as unparamet rized curve, is also an F 2 − geo d esic. W e sa y that they are affine e quivalent , if ev ery F 1 − geo d esic, considered as parametrized curve, is also an F 2 − geo d esic. Of course, in the d efinition we can replace any of the Finsler metrics by a Riemannian or p seudo-Riemannian one, or by an affine connection. Remark 1. Ge o desic e qu ivalenc e (or affine e qu ivalenc e) of Finsler metrics is not a priori a symmetric r elation, as Example 3 b elow shows. The r e ason is that for c ertain Finsler met- rics the uniq u eness the or em for the ge o desics do es not hol d: two differ ent ge o desics c an have the same ve lo city v e ctor, as in Example 3 b elow. Th en, even under the assumption that al l F 1 − ge o desics ar e F 2 − ge o desics, ther e may exist F 2 − ge o desics that ar e not F 1 − ge o desics. This phenomenon evidently do es not happ en, if the metrics ar e str ictly c onvex (and of c ourse in the Riemannia n c ase); for such metrics, F 1 is ge o desic al ly e quivalent to F 2 if and only if F 2 is ge o desic al ly e quivalent to F 1 . We wil l show in the b e g i nning of Se c tion 2.2.2 th at under the assump tion that the metric F is B e rw ald, if g is ge o desic al ly (or affine) e quivalent to F , then g is ge o desic al ly (o r affine, r esp.) e quiv alent to the c onne c tion Γ asso ciate d to F . possible geodesics Figure 1: The unit sphere in the norm p and p ossible geo desics of the corresp onding Minko ws ki metric 2 Example 3. Consid er the Minkowski metric F ( x, ξ ) = p ( ξ ) such that the unit spher e S 1 := { ξ ∈ R n | p ( ξ ) = 1 } is as on Figur e 1: the imp ortant fe atur e of the pictur e is tha t the p art of the unit sph er e lying in the marke d se ctor is a str aight line se gment. Then every c urve such that its velo city ve ctors ar e in the se ctor is a ge o desic. Besid e such curves, the str aight lines ar e also ge o desics. We se e that the standar d flat metric is ge o desic al ly and affine e quiv alent to F , but the metric F is neither ge o desic al ly nor affine e quivalent to the standar d fla t metric. Geod esic equiv alence of metrics is a classica l sub ject. Th e fir st non-trivial examples of geo desi- cally equiv alen t Riemannian metrics were d iscov ered b y Lagrange [La]. Geo desica lly equiv alent Riemannian metrics w ere studied b y Beltrami [Bel], Levi-Civita [LC], P ainlev ´ e [Pa] and other classics. One can find more historical details in the surveys [Am, Mi2] and in the intro d uction to the pap ers [Ma1, Ma4]. Geo desic equiv alence of Riemann ian and Finsler metrics is discu s sed in particular in Hilb ert’s 4th problem, see [Al1, P o]. Recen t results on geo d esic equiv alence of Riemannian and Finsler metrics includ e [MBB, Sh2]. Our main results are Theorem 1. L et F b e a Be rwald metric. Then ther e exists a Riemannian metric which is affine e qu ivalent to F . F or strictly con vex Finsler metrics, Theorem 1 is due to [Sz1]. Later, other pr oofs we re suggested in [Sz2, T o]. Our pro of is similar to the pro of in [Sz2]; the mo dification is b ased on the construction from [MR TZ]. Theorem 2. L et F b e an essential ly Berwald metric on a c onne cte d manifold, and let Γ b e its asso ciate d c onne ction. Supp ose a Riemannian or pseudo-R iemannian metric g is ge o desic al ly e quivalent to F , but is not affine e quivalent to F . Then ther e exists a c onstant µ , a symmetric (2 , 0) − tensor a ij , and a nonzer o ve ctor field λ i such that the fol lowing e quations ar e fulfil le d, wher e “ , ” denotes the c ovariant d erivative with r esp e ct to Γ : a ij ,k = λ i δ j k + λ j δ i k (1) λ i ,j = µ δ i j (2) W e see that equations (1 ,2 ) are of Cauc h y-F robeniu s t yp e, i.e., the d eriv ativ es of the unkn own functions a ij , λ i are explicitly expressed as fu nctio ns of the un kno wn functions and k n o wn data (connection Γ). Remark 2. If a Riema nnian metric g is affine e quiv alent to F , e quations (1,2) also have a nontrivial solution, namely a ij = g ij , λ i ≡ 0 , µ = 0 . Remark 3. The c onverse of The or em 2 is also true: the existenc e of a nonde gener ate a ij and of a nonzer o λ i satisfying e quations (1,2) for a c ertain c onstant µ implies the existenc e of a Riemannian or a pseudo-Riema nninan metric ge o desic al ly e quivalent to F , but not affine e quivalent to g . Recen tly , a system of Cauch y-F rob enius t yp e for metrics geodesically equiv alen t to Berw ald Finsler metrics w as obtained [MBB, Theorem 2]. Our system is muc h easier than one in 3 [MBB]: first of all, it is linear in the unkno wn fu nctio ns, second, it con tains less equations, and, third, the equations are muc h simpler than those of [MBB] and, in particular, con tain no curv ature terms. One cannot obtain our equations from the equations of [MBB ] by a c hange of unknown fu n ctio ns. In ord er to obtain our equations from those of [MBB], one should prolong the equations of [MBB] tw o times, and use the resu lt of the prolongation to simplify the system. Corollary 1. L et F b e an essential ly Berwald metric on a c onne cte d close d (= c omp act with- out b oundary) manifold. Then every Riema nnian or pseudo-Riema nnian metric ge o desic al ly e quivalent to F is affine e q uivalent to F . Corollary 2. L et F b e a c omplete essential ly Berwald metric on a c onne cte d manifold. Then every c omplete Riemannia n or pseudo-Riema nnian metric g e o desic al ly e quivalent to F is affine e quivalent to F . The assu mptions in Theorem 2 and Corollaries are im p ortant: it is p ossible to constru ct coun terexamples if the Berw ald metric is not essential ly Berw ald (i.e., is a Riemannian metric), or if one of the metrics is not complete. Corollary 3 (Hilb ert’s 4th problem for Berw ald metrics) . Supp ose an essential ly Berwald metric F on a c onne cte d manifold is pr oje ctively flat, that is, ther e exists a flat R iemannian metric ge o desic al ly e quivalent to F . Then F is isometric to a Minkowski metric. 2 Pro ofs 2.1 Av eraged metric and pro of of Theorem 1 Giv en a Finsler Berw ald m etric F , w e construct a Riemannian m et ric g = g F suc h that the asso cia ted conn ection Γ of F is the Levi-Civita connection of g implying that the m etric g is affine equiv alen t to F . As we mentio ned in the in tro duction, the constru ctio n is d ue to [MR TZ], and is similar to one from [Sz2]. Giv en a smo oth norm p on R n ≥ 2 , we canonically construct a p ositiv e d efinite symmetric bilinear form g : R n × R n → R . F or the Finsler metric F , the role of p will b e pla y ed by the restriction of F to T x M . W e w ill see that the constructed g smoothly dep ends on x , and hence it is a Riemannian metric. Consider the sph ere S 1 = { ξ ∈ R n | p ( ξ ) = 1 } . C onsider th e (un ique) vo lume form Ω on R n suc h that the v olume of th e 1-ball B 1 = { ξ ∈ R n | p ( ξ ) ≤ 1 } is equal to 1. Denote by ω the vol ume form on S 1 whose v alue on th e v ectors η 1 , ..., η n − 1 tangen t to S 1 at the p oin t ξ ∈ S 1 is giv en b y ω ( η 1 , ..., η n − 1 ) := Ω( ξ , η 1 , η 2 , ..., η n − 1 ). No w, for every p oin t ξ ∈ S 1 , consider the symmetric bilinear form b ( ξ ) : R n × R n → R , b ( ξ ) ( η , ν ) = D 2 ( ξ ) p 2 ( η , ν ). In this form ula, D 2 ( ξ ) p 2 is the s ec ond d ifferential at the p oin t ξ of the 4 function p 2 on R n . Th e analytic expression for b ( ξ ) in the co ordinates ( ξ 1 , ..., ξ n ) is b ( ξ ) ( η , ν ) = X i,j ∂ 2 p 2 ( ξ ) ∂ ξ i ∂ ξ j η i ν j . (3) Since the n orm p is con v ex, the bilinear f orm is n onnega tiv e defin ite. Clearly , f or ev ery ξ ∈ S 1 , w e ha ve b ( ξ ) ( ξ , ξ ) > 0 (4) (this is actually the reason why we tak e p 2 and not p in the definition of b ). No w consider the follo wing symmetric bilinear 2 − form g on R n : for η , ν ∈ R n , we p ut g ( η, ν ) = Z S 1 b ( ξ ) ( η , ν ) ω . (5) W e assume th at the orien tation of S 1 is chosen in suc h a wa y that R S 1 ω > 0. Because of (4 ), g is p ositiv e definite. No w let us extend this construction to eve ry tangent space T x M of the manifold, then F | T x M pla ys the role of p . S ince the construction dep ends smo ot hly on the p oint x ∈ M , w e ha v e that g := g F is a Riemann ian metric on M . W e sh o w that if the metric F is Berw ald with the asso cia ted connection Γ, then Γ is the Levi-Civita connection of g . Indeed, consider a smo oth cur v e γ connecting the p oints γ (0) , γ (1) ∈ M . Let τ : T γ (0) M → T γ (1) M b e the parallel transp ort of the v ectors along the curve with resp ect to the conn ec tion Γ. τ is a linear map. Since the metric is Berw ald, τ p reserv es the fun ctio n F and, in particular, the one-sphere S 1 . Since the forms Ω , ω w ere constru ct ed b y u sing the sphere S 1 and the linear structure of the sp ace only , τ pr eserv es the f orm ω . Since the function F is pr eserved as we ll, ev eryth ing in f ormula (5) is pr eserv ed b y the parallel tr ansp ort wh ich im p lies τ ∗ g = g . Th en g ij,k = 0, therefore ev ery (parametrized) geod esic of g is a geo desic of F . Theorem 1 is pro v ed . 2.2 Pro of of Theorem 2 and Corollaries 1, 2, 3 Within the w h ole section we assu me that our und erlying manifold is connected, orien table (otherwise w e pass to an orien table co ve r), and has dimension at least t wo. 2.2.1 Holonom y group of a Berw ald metric F Lemma 1. L et F b e an essential ly Berwald metric on a c onne cte d manifold M , and let g b e a R iemannian metric affine e quiv al ent to F (the existenc e of such metric is guar ante e d by The or em 1). Then, the metric g is symmetric of r ank ≥ 2 , or ther e exists one mor e R iemannian metric h su ch that it is not pr op ortional to g , but is affine e quivalent to g . 5 Pro of. W e essen tially rep eat the argumen tation of [S z1, Sz2]. T ak e a fi xed p oin t q ∈ M . F or ev ery (smo ot h) lo op γ ( t ) , t ∈ [0 , 1] w ith th e origin in q (i.e., γ (0) = γ (1) = q ), w e consider the parallel transp ort τ γ : T q M → T q M along the curv e. It is well kno wn (see for example, [Ber, Sim]), that the set H q := { τ γ | γ : [0 , 1] → M is a s m ooth lo op, γ (0) = γ (1) = q } is a su bgroup of the group of the orthogonal transformations of T q M . Moreo ver, it is also kno wn that at least one of the follo wing conditions holds: 1. H q acts transitiv ely on the unit sphere S 1 := { ξ ∈ T q M | g ( ξ , ξ ) = 1 } , 2. the metric g is symmetric of rank ≥ 2, 3. there exists one more Riemannian m etric h such that it is nonp rop ortional to g , but is affine equiv alen t to g . In the first case, since the holonomy group preserv es b oth g and F , the ratio F ( ξ ) 2 /g ( ξ , ξ ) is the same for all ξ ∈ T q M , ξ 6 = 0, implying that the metric g is Riemannian. L emma 1 is pro v ed. 2.2.2 Metrics with degree of mobilit y ≥ 3 If the dimension of the man if old is 2, an essen tially Berw ald metric is a Minko wski m et ric, and Theorem 2 and Corollaries 1, 2 , 3 are eviden t. Belo w, we assume th at the d im en sion of the manifold is ≥ 3. Supp ose t he (Riemannian or pseudo-Riemannian) metric ¯ g is geo desica lly equiv alen t to F , bu t is not affine equiv alen t to F . Then the metric ¯ g is geo desically equiv alen t to the a v eraged metric g = g F , but is not affine equiv alent to g . If th e u n iqueness theorem for geod esic s holds, the latter statemen t is trivial; for generic Finsler metrics, it p robably requires additional explanation. In order to explain wh y the metric ¯ g is geo desically equiv alent to the a verag ed metric g = g F , let us consider the set N := { ( x, ξ ) ∈ T M \ T M 0 | D 2 F 2 | T q M nondegenerate } . This set is eviden tly op en. As from th e follo wing standard (see for example [Ku]) argumen t from differen tial geometry it turns out, its inte rsection with ev ery T q M \ T M 0 is not empt y . W e need to show th at for a smo oth norm p := F | T q M on R n = T q M there exists a p oint s u c h that D 2 p 2 is nondegenerate at this p oin t. W e fix an Euclidean m et ric in R n and consider the sphere in R n (with resp ect to the c h osen Euclidean metric in T q M ) of large radiu s suc h th at the Finsler sphere S 1 := { ξ ∈ T q M | F ( ξ ) = 1 } lies inside, see the left-hand side of Figure 2. Then, w e mak e the radius smaller until the fi rst p oin t of the intersectio n of the sphere with S 1 , see the right-hand side of Fig ure 2. Clearly , at the p oin t of the int ersection, the second differen tial of p 2 is n ondegenerate as we claimed. 6 Figure 2: F or a smo oth norm p , there alwa y s exists a p oin t su c h that the sec ond differenti al of p 2 is n ondegenerate It is well known that for ( x, ξ ) ∈ N th e uniqu eness theorem of geo d esics holds: lo cally , th er e exists a u nique F − geo desic γ such that γ (0) = x and ˙ γ (0) = ξ . Moreo ver, the geo desic γ is also the geodesic of th e asso ciated connection Γ. T hen, ev ery ¯ g − geodesic su c h that ( γ (0) , ˙ γ (0 )) ∈ N is also a Γ − geo desic. Since the set N ∩ T q M is op en for eve ry q , th e connection ¯ Γ of ¯ g satifies the Levi-Civita condition Γ i j k − ¯ Γ i j k − 1 n +1  δ i k  Γ α j α − ¯ Γ α j α  + δ i j  Γ α k α − ¯ Γ α k α  = 0 at ev ery p oin t (in the pro of fr om [LC] it is sufficien t to assum e that only the geo d esics wh ose v elocit y v ectors are from certain op en set N ⊆ T M ; N ∩ T q M 6 = ∅ are common for b oth met- rics) implying that Γ and ¯ g are geo desical ly equiv alent , and hence g and ¯ g are also geo desically equiv alen t. Th us, the metric ¯ g is geo desicall y equiv alent to the a v eraged metric g as w ell, but n ot affine equiv alen t to g . By Lemma 1, the metric g is sym metric, or there exists a Riemannian metric h affine equiv alen t to g but not p rop ortio nal to g . W e sh o w that if the metric g is symmetric, the assumptions of Theorem 1 imp ly that it is flat from whic h it f ol lo ws th at there exists a metric h = h ij affine equiv alen t to g but not pr oportional to g at least on the unive rsal co ver of M , whic h is sufficien t for our goals. By a result of S injuk ov [Si1 ], ev ery symm et ric m etric geo desically equiv alen t to g is affine equiv alen t to g , unless the metric has constan t curv ature. In the latter case, th e m et ric must b e flat, otherwise the holonom y grou p d iscussed in the pr evio us section acts transitiv ely on the unit sphere, and the Finsler metric F is actually Riemannian. Th us, at least on the u niv ersal cov er of the manifold th er e exists a Riemannian metric h affine equiv alen t to g bu t not prop ortional to g . W e consid er the symmetric (1,1)-tensor a ij :=    det(¯ g ) det( g )    1 / ( n +1) ¯ g αβ g αi g β j , wher e ¯ g ij is the tensor, dual to ¯ g ij so that g iα ¯ g αj = δ j i , the f unction λ := 1 2 a αβ g αβ , and its differen tial λ i := ( dλ ) i := λ ,i . By the result of Sinjuko v [Si2], see also [BM] and [EM], if th e metric ¯ g is geo desically equiv alent to g , the tensor a ij and the (0,1) tensor λ i satisfy the equation a ij,k = λ i g j k + λ j g ik . (6) Moreo ve r, if the metrics g and ¯ g are n ot affine equiv alen t, λ i is n ot identic ally zero. Recall that th e de g r e e of mobility of the metric g is the dimens io n of the s p ace of solutions of equation (6) considered as equation on the u nkno wn a ij and λ i . In our case, the degree of 7 mobilit y is at least 3. In deed, ¯ a ij := g ij , ¯ λ i := 0 and ˆ a ij := h ij , ˆ λ i := 0 are also solutions, but b y th e assump tio ns they are linearly indep endent of th e solution a ij , λ i . Metrics w ith d egree of mobilit y ≥ 3 on manifolds of dimensions ≥ 3 were studied, in particular, in [KM], s ee also references th er ein. The last p art of the present pap er will essen tially use the results of [KM], so w e recommend the reader to ha v e [KM] at hand. By results of [K M , L emma 3], und er the ab o ve assu m ptions, f or eve ry solution a ij , λ i of equation (6), in a neigh b ourho o d of almost ev ery p oin t there exists a constan t B and a f unction µ suc h th at the follo wing equations h ol d: λ i,j = µ g ij + B a ij (7) µ ,i = 2 B λ i . (8) Indeed, equation (7) is equation (30) of [KM], and equation (8) is in [K M , Remark 8] (wh ere the function µ is denoted by ρ ). Our next goal is to sh o w that in our case B = 0 (and, therefore, equations (7) are fulfi lle d at ev ery p oint of the manifold, and the function µ is actually a constan t by (8)). T his will also imply that (6, 7) coincide with (1, 2) after raising indices w ith the help of g . In ord er to do th is, let us consider the solution A ij := a ij + h ij , Λ i := λ i + 0 = λ i , whic h is the sum of th e solutions a ij , λ i and h ij , 0. The data A ij , λ i satisfy equation (6). As we explained ab o ve , they therefore also satisfy equation (7) in a neigh b ourho o d of almost ev ery p oin t, i.e., in a n eig h b ourho od of almost ev ery p oint there exist a fu nction ˜ µ and a constan t ˜ B such that λ i,j = ˜ µg ij + ˜ B ( a ij + h ij ) . (9) Subtracting equation (7) from (9), w e obtain ( µ − ˜ µ ) g ij = ( ˜ B − B ) a ij + ˜ B h ij . (10) W e see that the righ t-hand side of equation (10) is a linear com b inatio n of t w o solution a ij and h ij and is therefore also a solution of (6) (w ith an appropriate λ i ). As it w as prov ed in [BKM, Lemma 1] (the result is essen tially due to W eyl [W e]), the function µ − ˜ µ must b e a constan t. Since g , a, and h are linearly indep endent , all co efficien ts in th e linear com b in at ion (10) are zero implying B = 0. Th us, equations (6, 7) coincide w ith equations (1,2) after raising the indexes. Theorem 2 is pro v ed. Pro of of Corollaries 1,2. As we explained ab o ve, w e can assu me that th e dimension of the manifold is ≥ 3 and the degree of mobility is ≥ 3. Und er th ese assumptions, Corollary 1 follo ws from [KM, Theorem 2] (if g is Riemannian, the result is du e to [Ma4, Th eorem 16]; in view of Theorem 2 , the r esult follo ws from [Mi1, Theorem 5]), and Corollary 1 follo ws from [Ma3, Theorem 2] (if g is Riemannian, the result is du e to [KM, Theorem 1]). Pro of of Corollary 3. S upp ose that a fl at Riemannian metric ¯ g is geod esica lly equiv alen t to an essen tially Berw ald metric F . Consider the av eraged metric g = g F constructed in Section 8 2.1. It is affine equiv alent to F , and , therefore, as we explained in Section 2.2.2, is geo desically equiv alen t to ¯ g . By the classical Beltrami T heorem (see for example [Ma2], or the original pap ers [Bel] and [Sc]), the metric g has constant curv atur e. I f the curv ature of g is not zero, the h olo non y group of g acts transitivel y on the un it sphere implying the metric F is actually Riemann ia n. Th us, the metric g is flat. Then, there exists a co ordinate system suc h that Γ ≡ 0. In this co ordinate sys te m, paralle l trans p ort along a cu rv e d o es not dep end on the cu rv e and is the usual p aral lel transation x 7→ x + T . Since the parallel tr an s port p reserv es F , we ha ve that F is tr anslat ion-in v arian t implying it is Minko wski metric as we claimed. Ac knowledgemen ts. I thank Deutsche F orsc hungsgemeinsc haft (Priorit y Program 1154 — Global Differen tial Geometry) and FSU Jena for p artia l financial supp ort, and V. Kiosak for attract ing my a tten tion to the pap er [MBB] and for useful discussions. T he idea to use the construction from [MR TZ] in the pro of of Theorem 1 app eared d u ring the lunch with L. Kozma and T. Q. Binh; I thank th em and the Universit y of Debrecen for their hospitalit y . I also thank J. Mik es and Z. Shen f or their commen ts, and the r eferee f or grammatical and st ylistic corrections. References [Al1] J . C. ´ Alv arez P aiv a: Symple ctic ge ometry and Hilb ert’s fourth pr oblem. J. Differentia l Geom. 69 (2005) , no. 2, 353–37 8. [Al2] J . C. ´ Alv arez P aiv a: Some pr oblems on Finsler ge ometry. Handb ook of differen tial ge- ometry . V ol. I I, 1–33, Elsevier/North-Holland, Amsterdam, 2006. [Am] A. V. Aminov a: Pseudo-R iemannian manifolds with gener al ge o desics. Ru ssian Math. Surveys 48 (1993), no. 2, 105–160 , MR1239 862, Z bl 0933.530 02. [BCS] D. Bao, S.- S. Ch ern, Z. Shen: An Intr o duction to R iemann-Finsler Ge ometry. Grad. 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