A class of Finsler metrics admitting first integrals
aa r X i v : . [ m a t h . DG ] J a n A CLASS OF FINSLER METRICS ADMITTING FIRST INTEGRALS
IOAN BUCATARU, OANA CONSTANTINESCU, AND GEORGETA CRET¸ U
Abstract.
We use two non-Riemannian curvature tensors, the χ -curvature and the mean Berwaldcurvature to characterise a class of Finsler metrics admitting first integrals. Introduction
Finsler geometry is a natural extension of Riemannian geometry and, while many geometricstructures can be extended from the Riemannian to the Finslerian setting, within the Finsleriancontext there are many non-Riemannian geometric quantities, [17, Ch. 6].Existence of first integrals is of great importance, they provide a lot of information about thecorresponding geometry, including some rigidity results, [6], [7], [21]In Riemannian geometry, Topalov and Matveev obtained in [21], for two projectively equivalentmetrics on an n -dimensional manifold, a set of n first integrals. An extension of this result to theFinslerian context has been proposed by Sarlet in [16]. In [7], Foulon and Ruggiero have shownthe existence of a first integral for k -basic (of isotropic curvature) Finsler surfaces.It has been proven by Li and Shen in [11], that Finsler metrics of isotropic curvature can becharacterised using the χ -curvature tensor. The χ -curvature has been introduced by Shen in [18],in terms of another important non-Riemannian quantity, the Shen-function ( S -function) [17, § χ -curvature, [10, 14, 19].In this work we extend the result of Foulon and Ruggiero from [7] to Finsler manifolds ofarbitrary dimension, by providing a class of Finsler metrics that admit first integrals. This classof Finsler metrics is characterised using the χ -curvature tensor and the mean Berwald curvature, E jk = B iijk , where B ijkl is the Berwald curvature, [17, § S -function. The S -function is a Finsler function if and only if the mean Berwald curvature has maximal possiblerank, n −
1. For a Finsler function F , we denote by det g , the determinant of its metric tensor g ij = ∂ F ∂y i ∂y j , where y i are the fiber coordinates in the tangent bundle T M .The main result of this paper provides a class of Finsler metrics that admit a first integral.
Theorem 1.1.
Consider F a Finsler metric that satisfies the following two conditions: i) the χ -curvature vanishes; ii) the mean Berwald curvature has rank n − . Date : January 29, 2021.2000
Mathematics Subject Classification.
Key words and phrases.
Finsler metric, χ -curvature, scalar mean Berwald curvature, first integral. Then, λ = − g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F E ij ∂F∂y i ∂F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1.1) is a first integral for the geodesic spray G of the Finsler metric F , which means that G ( λ ) = 0 . For a Finsler surface, the first condition of Theorem 1.1, χ = 0, is equivalent to the fact that theFinsler metric has isotropic curvature (it is a k -basic Finsler metric). Also, in dimension 2, the meanBerwald curvature is proportional to the vertical Hessian of the Finsler metric, the proportionalityfactor, the function λ , was known since Berwald, [1, (8.7)]. Hence for Finsler surfaces, the secondcondition of Theorem 1.1 is automatically satisfied. Moreover, the first integral λ , given by formula(1.1), reduces in the 2-dimensional case to the first integral f obtained by Foulon and Ruggiero in[7, Theorem B].For the proof of Theorem 1.1, the two conditions χ = 0 and rank( E ij ) = n − S -function is a Finsler metric, projectively related to F . Then, we will obtain the first integral λ ,given by (1.1), using the Painlev´e first integral, associated to the two projectively related Finslermetrics F and S .Next theorem deals with a concrete class of Finsler metrics that satisfy the second assumptionof Theorem 1.1. We say that a Finsler metric F has scalar mean Berwald curvature f if the meanBerwald curvature is proportional to the vertical Hessian of F , 2 E ij = f F y i y j . Theorem 1.2.
Consider F a Finsler metric that satisfies the following two conditions: i) the χ -curvature vanishes; ii) the Finsler metric has scalar mean Berwald curvature f .Then, the scalar mean Berwald curvature satisfies: f is a first integral of the Finsler metric F . If dim M > then the first integral f is constant. If M is compact and dim M > then the first integral f vanishes identically. The proof of Theorem 1.2 is a direct extension, to the n -dimensional case, of the techniques usedby Foulon and Ruggiero in [7] to prove the existence of a first integral for k -basic Finsler surfaces.These techniques allow to provide more information about the first integral and one can furtheruse [6] to obtain a rigidity result for the class of Finsler metrics with vanishing χ -curvature andscalar mean Berwald curvature.2. Finsler metrics: a geometric setting and some non-Riemannian quantities
In this work, we assume that M is an n -dimensional C ∞ - manifold, of dimension n >
1. Weconsider
T M its tangent bundle and T M = T M \ { } the tangent bundle with the zero sectionremoved. Local coordinates on M , denoted by ( x i ), induce canonical coordinates on T M (and T M ), denoted by ( x i , y i ). On T M there are two canonical structures that we will use: theLiouville (dilation) vector field, C = y i ∂∂y i , and the tangent structure (vertical endomorphism), J = ∂∂y i ⊗ dx i .2.1. A geometric setting for Finsler metrics.
We will use the Fr¨olicker-Nijenhuis theory todescribe the geometric setting we follow in this work. For a vector-valued l -form L on T M , wedenote by i L the induced i ∗ -derivation of degree ( l −
1) and by d L := [ i L , d ] the d ∗ derivation of CLASS OF FINSLER METRICS ADMITTING FIRST INTEGRALS 3 degree l , [3, 8, 9, 20, 22]. For two vector valued forms, an l -form L and a k -form K , consider the( k + l )-form [ K, L ], uniquely determined by d [ K,L ] = d K d L − ( − kl d L d K . A spray is a second order vector field G ∈ X ( T M ) such that JG = C and [ C , G ] = G . Locally,a spray can be expressed as G = y i ∂∂x i − G i ∂∂y i , with the functions G i ( x, y ) positively 2-homogeneous in y (2 + -homogeneous). A geodesic of a spray G is a smooth curve c on M whose velocity ˙ c is an integral curve of G , G ( ˙ c ( t )) = ¨ c ( t ).For a given spray G , an orientation preserving reparameterization t → ˜ t ( t ), of its geodesics,leads to a new spray e G = G − P C , where P is a 1 + -homogeneous function on T M . We say thatthe two sprays G and e G are projectively related , while P is called the projective factor . Definition 2.1. A Finsler structure on M is a continuous function F : T M → [0 , + ∞ ) thatsatisfies the following conditions:i) F is smooth on T M ;ii) F is 1 + -homogenous, F ( x, λy ) = λF ( x, y ), ∀ λ > ∀ ( x, y ) ∈ T M ;iii) the metric tensor g ij = 12 ∂ F ∂y i ∂y j is non-degenerate on T M .A Finsler manifold is a pair ( M, F ), with F a Finsler structure on the manifold M . Fora Finsler manifold, one can identify the sphere bundle SM with the indicatrix bundle IM = { ( x, y ) ∈ T M, F ( x, y ) = 1 } . Geometric objects on T M that are invariant under positive rescaling(0 + -homogeneous) can be restricted to the sphere bundle SM .For a Finsler structure F , the metric tensor g ij can be expressed in terms of the angular metric h ij as follows: g ij = h ij + 1 F y i y j = h ij + ∂F∂y i ∂F∂y j , h ij = F ∂ F∂y i ∂y j = F F y i y j , where y i = g ik y k = F ∂F∂y i . The regularity condition iii) from Definition 2.1 is equivalent to the factthat the angular metric h ij has rank n −
1, [12, Proposition 16.2].For a spray G and a function L on T M , we consider the Euler-Lagrange δ G L := L G d J L − dL = (cid:26) G (cid:18) ∂L∂y i (cid:19) − ∂L∂x i (cid:27) dx i . (2.1)Every Finsler metric uniquely determines a geodesics spray , solution of the Euler-Lagrange equation δ G F = 0.We recall now the geometric structures induced by a Finsler metric, and its geodesic spray G . We first have the canonical nonlinear connection , characterised by a horizontal and a verticalprojector on T M h = 12 (Id − [ G, J ]) , v = 12 (Id +[ G, J ]) . In induced local charts on T M , the two projectors can be expressed as: h = δδx i ⊗ dx i , v = ∂∂y i ⊗ δy i , where δδx i = ∂∂x i − N ji ∂∂y j , δy i = dy i + N ij dx j and N ij = ∂G i ∂y j . CLASS OF FINSLER METRICS ADMITTING FIRST INTEGRALS 4
Lemma 2.2.
Consider F a Finsler metric and e F a + -homogeneous function, nowhere vanishingon T M . Then, we can express the determinant of the metric tensor g ij as follows: det g = − F n +1 e F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ F∂y i ∂y j ∂ e F∂y i ∂ e F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.2) Proof.
First, we recall a formula that connects the determinant of the metric tensor g ij in termsof the angular metric h ij , [15, (1.26)]:det g = − F n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ F∂y i ∂y j ∂F∂y i ∂F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.3)For the metric tensor g ij , consider { h = GF , h , ..., h n } an orthonormal horizontal basis and { h i , i = 1 , n } , the dual frame. Since, for α ≥ h α ( h ) = 0, and h α = h αi dx i , h = y i F δδx i , we obtain h αi y i = 0. Using also h ij y j = 0, we obtain, for each α ≥
2, that on T M , (cid:18) h ij h αi h αj (cid:19) y ... y n = 0and consequently, (cid:12)(cid:12)(cid:12)(cid:12) h ij h αi h αj (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (2.4)The semi-basic 1-form d J e F can be expressed as follows d J e F = ∂ e F∂y i dx i = d J e F ( h i ) h i = e FF h + n X α =2 d J e F ( h α ) h α = ( e FF ∂F∂y i + n X α =2 J ( h α )( e F ) h αi ) dx i . CLASS OF FINSLER METRICS ADMITTING FIRST INTEGRALS 5
In the determinant from (2.3), we replace e FF ∂F∂y i and obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ F∂y i ∂y j ∂F∂y i ∂F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) F e F (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ F∂y i ∂y j e FF ∂F∂y i e FF ∂F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) F e F (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ F∂y i ∂y j ∂ e F∂y i ∂ e F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − n X α =2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ F∂y i ∂y j J ( h α )( e F ) h αi J ( h α )( e F ) h αi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.4) = (cid:18) F e F (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ F∂y i ∂y j ∂ e F∂y i ∂ e F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We replace this in formula (2.3) to obtain (2.2) and complete the proof. (cid:3)
For a Finsler metric F , the regularity condition iii) of Definition 2.1 can be reformulated interms of the Hilbert 1-form d J F . Since d J F is 0 + -homogeneous, we can view it as a 1-form on SM . Due to the fact that the Hilbert 2-form can be expressed as dd J F = ∂ F∂y i ∂y j δy j ∧ dx i = 1 F h ij δy j ∧ dx i , (2.5)it follows that d J F is a contact structure on SM and hence the (2 n − ω SM = d J F ∧ dd J F ( n − is a volume form on SM .2.2. Non-Riemannian structures in the Finslerian setting.
The first non-Riemannian struc-tures, associated to a Finsler metric F , are the Cartan torsion and the mean Cartan torsion , C ijk = 14 ∂ F ∂y i ∂y j ∂y k = 12 ∂g ij ∂y k , I k = g ij C ijk . A Finsler metric reduces to a Riemannian metric if and only if the (mean) Cartan torsion vanishes.The curvature of the nonlinear connection determined by the geodesic spray G is defined by R = 12 [ h, h ] = 12 R ijk ∂∂y i ⊗ dx j ∧ dx k = 12 δN ij δx k − δN ik δx j ! ∂∂y i ⊗ dx j ∧ dx k . The canonical nonlinear connection provides a tensor derivation on T M , the dynamical covari-ant derivative ∇ , whose action on functions and vector fields is given by [3, (21)]: ∇ ( f ) = G ( f ) , ∀ f ∈ C ∞ ( T M ) , ∇ X = h [ G, hX ] + v [ G, vX ] , ∀ X ∈ X ( T M ) . The geodesic spray G of a Finsler metric induces a linear connection on T M , the Berwald con-nection, [20, § Berwald curvature and the
Riemanniancurvature , [17, § § , 8.1]: B ijkl = ∂ G i ∂y j ∂y k ∂y l , R ijkl = ∂R ikl ∂y j . The mean Berwald curvature of a spray G is defined as [17, Def. 6.1.2] E jk = 12 B iijk = 12 ∂ G i ∂y i ∂y j ∂y k . (2.6) CLASS OF FINSLER METRICS ADMITTING FIRST INTEGRALS 6
Definition 2.3.
A Finsler metric has scalar mean Berwald curvature if the mean Berwald cur-vature is proportional to the angular metric, which means that there exists a 0 + -homogeneousfunction f on T M such that E ij = 12 fF h ij = f ∂ F∂y i ∂y j . (2.7)In [5], Chen and Shen study Finsler metrics of isotropic mean Berwald curvature , with a similardefinition as above, with f being a scalar function on M .In the next lemma we prove that in dimension n >
2, Finsler metrics of scalar mean Berwald cur-vature have isotropic mean Berwald curvature. In other words, the scalar mean Berwald curvature f is constant along the fibres of T M . Lemma 2.4.
Consider F a Finsler metric of scalar mean Berwald curvature f . If n > , then f is constant along the fibres of T M .Proof. From the definition of the mean Berwald curvature (2.6) we obtain that its vertical derivativeis a (0 , scalarmean Berwald curvature we have ∂E ij ∂y k = ∂E ik ∂y j (2.7) = ⇒ ∂f∂y k ∂ F∂y i ∂y j = ∂f∂y j ∂ F∂y i ∂y k = ⇒ ∂f∂y k h ij = ∂f∂y j h ik = ⇒ ∂f∂y k (cid:18) g ij − F y i y j (cid:19) = ∂f∂y j (cid:18) g ik − F y i y k (cid:19) . In the last formula above, we multiply with g il , the inverse of the metric tensor and obtain: ∂f∂y k (cid:18) δ lj − F y l y j (cid:19) = ∂f∂y j (cid:18) δ lk − F y l y k (cid:19) . Now, if we use that f is 0 + -homogeneous and take the trace, j = l , we obtain ( n − ∂f /∂y k = 0.Since n >
2, we obtain that the function f is constant along the fibres of T M . (cid:3) Due to the 2 + -homogeneity of the spray coefficients G i , it follows that E ij y j = 0, hencerank( E ij ) ≤ n −
1. In the 2-dimensional case, we obtain that the mean Berwald curvature hasrank 1, it is proportional to the angular metric h ij (of rank 1 as well), and hence all 2-dimensionalFinsler manifolds have scalar mean Berwald curvature. The proportionality factor has been knownsince Berwald, [1, (8.7)], but it has been shown only recently that it is a first integral for k -basicFinsler surfaces, [7, Theorem B].The Berwald connection is not a metric connection, with respect to the metric tensor of a Finslerstructure. Due to this non-metricity property of the Berwald connection, it follows that the (0 , R ijkl = g is R sjkl is not skew-symmetric in the first two indices, [17,(10.6)], and hence R iikl = 0. A measure of this failure is given by the χ -curvature, [19, Lemma 3.1]: χ j = − R iijk y k . This non-Riemannian quantity has been introduced by Shen in [18].The key ingredients we will use in this work are the χ -curvature, the mean Berwald curvature,and the fact that both curvature tensors can be expressed in terms of yet another non-Riemannianquantity, the S -function. CLASS OF FINSLER METRICS ADMITTING FIRST INTEGRALS 7
For a fixed vertically invariant volume form σ ( x ) dx ∧ dx ∧ · · · ∧ dx n ∧ dy ∧ dy ∧ · · · ∧ dy n on T M , [20, p. 490], we consider the Shen-function ( S - function ) and the distortion τ , [17, § S = G ( τ ) , τ = 12 ln det gσ . (2.8)From the various expressions of the χ -curvature, we will use its expression in terms of the S -function, [18, (1.10)], χ = 12 δ G S = 12 (cid:26) ∇ (cid:18) ∂S∂y i (cid:19) − δSδx i (cid:27) dx i . (2.9)The mean Berwald curvature can also be expressed in terms of the S -function as follows, [17,(6.13)]: E ij = 12 ∂ S∂y i ∂y j . (2.10)In view of formula (2.10), the second assumption of Theorem 1.1 or 1.2, assures that the verticalHessian of the S -function has maximal rank ( n − S -functionas a Finsler metric on its own. 3. Proof of Theorem 1.1
For the proof of Theorem 1.1, we proceed with the following steps. We show first that the twoassumptions of Theorem 1.1 assure that the S -function is a Finsler metric, projectively related to F . Then, we obtain the first integral (1.1) using the Painlev´e first integral associated to the twoprojectively related Finsler metrics S and F .Two Finsler metrics F and e F are projectively related if their geodesic sprays G and e G areprojectively related. One can characterise projective equivalence of two Finsler metrics F and e F using the following equivalent forms of Rapcs´ak equations, [20, § R ) δ G e F = 0;( R ) d h d J e F = 0.In Riemannian geometry, Topalov and Matveev [21, Theorem 1] associate to each pair of geodesi-cally equivalent metrics a set of n first integrals. An extension of this result, to the Finsleriansetting, has been proposed by Sarlet in [16] and his Ph.D student Vermeire [23].In the next lemma, we show that two projectively related Finsler metrics F and e F induce afirst integral (Painlev´e first integral). This first integral, given by formula (3.1), is the Finslerianextension of the first integral determined by two projectively equivalent Riemannian metrics, [13,Theorem 2]. Lemma 3.1.
Consider F and e F , two projectively related Finsler metrics. Then, I = e FF (cid:18) det g det e g (cid:19) n +1 (3.1) is a first integral for F .Proof. For a Finsler metric F , the dynamical covariant derivative of its metric tensor vanishes, [2],hence: ∇ ( g ij ) = G ( g ij ) − g im N mj − g mj N mi = 0 . CLASS OF FINSLER METRICS ADMITTING FIRST INTEGRALS 8
Contracting with g ij , we obtain g ij G ( g ij ) = g ij ( g im N mj + g mj N ji ) = 2 N ii , and hence N ii = 12 G (ln(det g )) . The two Finsler metrics F and e F being projectively related, their geodesic sprays and nonlinearconnections are connected through e G = G − P C , f G i = G i + P y i , f N ij = N ij + ∂P∂y j y i + P δ ij . If in the last formula above we take the trace i = j , it follows that the projective factor P is givenby P = 1 n + 1 ( e N ii − N ii ) = 12( n + 1) G (cid:18) ln (cid:18) det e g det g (cid:19)(cid:19) . We also use the alternative expression of the projective factor P , P = G ( e F )2 e F = 12 G (cid:16) ln e F (cid:17) . By comparing the two expressions of the projective factor P , we obtain G ( I ) = 0, which concludesthe proof of our lemma. (cid:3) We will give the proof of Theorem 1.1 now. The second assumption ii) on Theorem 1.1 togetherwith formula (2.10) assure that the angular metric of the S -function has rank n − S is a Finsler metric. The vanishing of the χ -curvature (2.9) assures that the Finsler metric S isprojectively related to F . In view of Lemma 3.1 we obtain that I = SF (cid:18) det g det s (cid:19) n +1 is a first integral for the Finsler metric F .We will use Lemma 2.2 for the Finsler metric S and the 1 + -homogenous function F . Accordingto formula (2.2), we can express the determinant of the metric tensor s ij as follows:det s = − S n +1 F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ S∂y i ∂y j ∂F∂y i ∂F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.10) = − S n +1 F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ij ∂F∂y i ∂F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − S n +1 F n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F E ij ∂F∂y i ∂F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .. Since I is a first integral for the Finsler metric F , it follows that1 I n +10 = F n +1 S n +1 det s det g = − g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F E ij ∂F∂y i ∂F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is also a first integral for F that coincides with λ given by formula (1.1).The first two assumptions of Theorem 1.1 tell us that S is a Finsler metric projectively relatedto F . One can use this and [21, Theorem 1] and [16, Theorem 2] to provide a set of n first integralsfor Finsler metric with vanishing χ -curvature and mean Berwald curvature of maximal rank. CLASS OF FINSLER METRICS ADMITTING FIRST INTEGRALS 9 Proof of Theorem 1.2
Partial proof of Theorem 1.2.
First we prove the first two conclusions of Theorem 1.2,using Theorem 1.1. For this proof it is essential that the scalar mean Berwald curvature f is nowherevanishing, hence we cannot reach the third conclusion of Theorem 1.2 using these techniques.In view of the equivalence of the two Rapcs´ak equations R and R , we can reformulate thevanishing of the χ -curvature (2.9) as d h d J S = 0. Using also the assumption that F has scalarmean Berwald curvature, we obtain that the Hilbert 2-form of the S -function can be written asfollows dd J S = d v d J S = ∂ S∂y i ∂y j δy i ∧ dx j = 2 E ij δy i ∧ dx j = f ∂ F∂y i ∂y j δy i ∧ dx j = f dd J F. (4.1)For a non-vanishing scalar mean Berwald curvature f , it follows from (4.1) that rank (cid:18) ∂ S∂y i ∂y j (cid:19) = n − S is a Finsler metric.We will express now, the first integral λ , (1.1), using the assumption that F has scalar meanBerwald curvature f . We have λ = − g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F E ij ∂F∂y i ∂F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − F n − det g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ∂ F∂y i ∂y j ∂F∂y i ∂F∂y j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.3) = f n − . (4.2)Since λ is a first integral, it follows that f is also a first integral for F , and this proves the firstconclusion of Theorem 1.2.According to Lemma 2.4, the scalar mean Berwald curvature f is a scalar function on M , whichmeans that d J f = 0. We use now that G ( f ) = 0, which can be written as ∇ f = 0. If we apply d J to this formula and use the commutation rule for ∇ and d J , [4, (2.11)], we obtain0 = d J ∇ f = ∇ d J f + d h f. Therefore, d h f = 0 and hence f is a constant, which proves the second conclusion of Theorem 1.2.4.2. Complete proof of Theorem 1.2.
In this section we present a proof of Theorem 1.2,independent of the results of Theorem 1.1, by extending to the n -dimensional case, the techniquesof [7]. This method allows to provide more information about the first integral, when the basemanifold is compact.The mean Cartan torsion can be expressed in terms of the distortion τ , and it does not dependon the fixed volume form on M , I k = 12 g ij ∂g ij ∂y k = ∂∂y k (ln p det g ) = ∂τ∂y k , I = I k dx k = d J (ln p det g ) = d J τ. The key ingredient in this proof is the following 1-form α = i [ J,G ] L G I = ∇ I k dx k − I k δy k = ∇ d J τ − d v τ (4.3) = d J ∇ τ − d h τ − d v τ = d J ∇ τ − dτ = d J S − dτ. In the 2-dimensional case, this form reduces to the form α from [7, § α to connect it with the χ -curvature: L G α = L G d J S − L G dτ = L G d J S − dS = δ G S = 2 χ. (4.4) CLASS OF FINSLER METRICS ADMITTING FIRST INTEGRALS 10
In view of this formula, the χ -curvature vanishes if and only if the form α is invariant by thegeodesic flow. Moreover, the χ -curvature vanishes if and only if the S -function satisfies the Rapcs´akequation δ G S = 0, which is equivalent to d h d J S = 0.Therefore, we can express the 2-form dα as follows: dα = dd J S = d h d J S + d v d J S = ∂ S∂y i ∂y j δy i ∧ dx j = 2 E ij δy i ∧ dx j . If we consider now the assumption that the Finsler metric has scalar mean Berwald curvature,then the 2-form dα is proportional to the Hilbert 2-form dd J F : dα = 2 E ij δy i ∧ dx j = fF h ij δy i ∧ dx j = f dd J F. (4.5)From formula (4.4) we obtain that χ = 0 implies L G α = 0 and therefore L G dα = 0. In viewof formula (4.5) and using the fact that L G dd J F = 0 we obtain G ( f ) = 0, which means that thescalar mean Berwald curvature f is a first integral for the geodesic flow G .Using Lemma 2.4 we obtain that the scalar mean Berwald curvature f is a scalar function on M , hence df = d h f . From formula (4.5), we obtain0 = d α = df ∧ dd J F = d h f ∧ d v d J F = 12 (cid:18) ∂f∂x i h kj − ∂f∂x j h ki (cid:19) dx i ∧ dx j ∧ δy k . It follows that ∂f∂x i h kj = ∂f∂x j h ki = ⇒ ∂f∂x i (cid:18) g kj − F y k y j (cid:19) = ∂f∂x j (cid:18) g ki − F y k y i (cid:19) . In the last formula above, we multiply with g il and obtain: ∂f∂x i (cid:18) δ lj − F y l y j (cid:19) = ∂f∂x j (cid:18) δ li − F y l ky i (cid:19) . If we take the trace l = j , we obtain( n − ∂f∂x i = − F G ( f ) y i . Now using that G ( f ) = 0, we obtain that the scalar function f is constant if dim M > f and can be useful for some rigidity results. Lemma 4.1.
Let ( M, F ) be a compact Finsler manifold with vanishing χ -curvature and of scalarmean Berwald curvature f . Then, Z SM f ω SM = 0 . (4.6) Proof.
By Stokes Theorem we have that0 = Z SM d (cid:0) α ∧ d J F ∧ ( dd J F ) n − (cid:1) = Z SM dα ∧ d J F ∧ ( dd J F ) n − − Z SM α ∧ ( dd J F ) n − . We will prove now that on SM , α ∧ ( dd J F ) n − = 0.Let λ , .., λ n − be the non-zero eigenvalues of the angular metric h ij , h , ..., h n − the corre-sponding horizontal eigenvectors and v i = Jh i , i ∈ { , ..., n − } . Then, { h , ..., h n − , v , ..., v n − } is a local frame of the (2 n − d J F ) on SM . We consider also the CLASS OF FINSLER METRICS ADMITTING FIRST INTEGRALS 11 local co-frame { h , ..., h n − , v , ..., v n − } . Using the expression (2.5) of the Hilbert 2-form, dd J F ,it follows that ( dd J F ) n − = λ · · · λ n − h ∧ · · · h n − ∧ v · · · ∧ v n − . Since i G α = 0, it follows that α ∈ span { h , ..., h n − , v , ..., v n − } = Ker( d J F ), we obtain that α ∧ ( dd J F ) n − = 0 on SM . Now, using (4.5), we obtain0 = Z SM dα ∧ d J F ∧ ( dd J F ) n − = Z SM f dd J F ∧ d J F ∧ ( dd J F ) n − = Z SM f ω SM . (cid:3) If dim
M > f is constant and using formula (4.6) we obtain that f = 0, which completesthe proof of Theorem 1.2.Existence of first integrals for Finsler manifolds can be used to provide rigidity results undersome topological restrictions: • compact surface, without conjugate points and of genus greater than one, [7, Theorem A]; • compact manifold, without conjugate points and of uniform visibility, for dimension n > M is a compact manifold of dimension n >
2, with vanishing χ -curvature and of scalar meanBerwald curvature f , we obtain that f = 0. Using formula (4.5), it follows that the form α , givenby (4.3), is closed. Using the assumptions of [6, Theorem A] we can conclude that the form α isexact. Assume α = dh , for some function h on T M . Since i G α = 0, it follows that G ( h ) = 0 and h is a first integral for the geodesic flow. Using again [6, Theorem A] we obtain that the function h is constant, then α = 0. The expression (4.3) of the form α allows to conclude that the meanCartan tensor vanishes, I = 0, and hence ( M, F ) is a Riemannian manifold.
Acknowledgments.
We express our thanks to J´ozsef Szilasi for his comments and suggestionson this work.
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