Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 25 (2009), ISSN

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1 Acta Mathematica Academiae Paedagogicae Nyíregyháziensis ), ISSN PROJECTIVE RANDERS CHANGES OF SPECIAL FINSLER SPACES S. BÁCSÓ AND Z. KOVÁCS Abstract. A change of Finsler metric Lx, y) Lx, y), is called a Randers change of L if Lx, y) = Lx, y) + b α x)y α. The purpose of this paper is to study the conditions for a Finsler space of weakly Berwald/Landsberg type which could be transformed by a Randers change to a Finsler space of the same type.. Introduction Randers s well-known method for giving examples of Finsler spaces has the form Lx, y) = a ij x)y i y j + b i x)y i where a ij is a Riemannian metric and βy i ) = b i y i is a one form with the condition b = a ij b i b j < a ij is the inverse of a ij ). If we change αx, y) = aij x)y i y j to a given Finsler metric, this method may lead to another Finsler metric. Definition [5]). A change of Finsler metric Lx, y) Lx, y), is called a Randers change of L if ) Lx, y) = Lx, y) + bi x)y i where βx, y) = b i x)y i is a one form on a smooth manifold M n. Thorough this paper we always suppose the regularity, positive homogeneity and strong convexity for the Finsler structure [3]), thus we assume a priory that L satisfies the ordinary conditions as fundamental function. Another important change of Finsler metrics is the so called projective change. A change of Finsler metric Lx, y) Lx, y), is called a projective change of L 2000 Mathematics Subject Classification. 53B40. Key words and phrases. Randers change, weakly-berwald space, Landsberg space. 65

2 66 S. BÁCSÓ AND Z. KOVÁCS if geodesic curves are preserved. It is a well-known fact that Lx, y) Lx, y) is projective if and only if there exists a scalar field px, y) which is positive homogeneous of order one, called the projective factor, satisfying Ḡi = G i + px, y)y i where G i are the geodesic spray coefficients. Projective Randers changes are characterised by the following theorem: Theorem [4]). A Randers change is projective if and only if b is a gradient vector field. Randers changes of special Finsler spaces were studied e.g. in the papers [], [7]. In [7] Park and Lee gave conditions for Finsler spaces changed by a Randers change to be of Douglas type. Theorem [7]). Let F n M n, L) F n M n, L) a projective Randers change. If F n is a Douglas space, then F n is also a Douglas space, and vice versa. The terminology and notations are referred basically to monograph [6]. Let M n be an n-dimensional n > 2) differentiable manifold and F n be a Finsler space equipped with a fundamental function Lx, y) on M n. A short review of the basic notations: the Finsler metric tensor: g ij = i j L 2 /2 where i refers to the partial derivation with respect to y i. g ij is the inverse of g ij the distinguished section: l i = y i /L, l i = y i /L the angular metric tensor: h ij = g ij l i l j the geodesic spray coefficients and successive y-derivatives: the hv-torsion 4G j = j i L 2 )y i j L 2, G i = g iα G α, G i j = j G i, G i jk = k G i j, Gi jkl = l G i jk, g αl G α ijk = G lijk 2) 2P ijk = y α G α ijk. Throughout the paper we shall use the notation L i = i L, L ij = j i L etc. We use the following properties of the angular metric tensor freely: h ij = LL ij h ij l j = 0 g ij h ik = δ j k lj l k g ij h ij = n. In the projective geometry of Finsler manifolds, there is an important projective invariant quantity, the Douglas tensor defined by 3) Dijk h = Gh ijk Gijk y h + δi h n + G jk + δj h G ik + δk h G ) ij.

3 PROJECTIVE RANDERS CHANGES OF SPECIAL FINSLER SPACES Projective Randers changes Lemma. For a Randers change we have L h ij = L h ij. Proof. It follows from ) that L i = L i + b i, Lij = L ij. The angular metric tensor satisfies h ij = LL ij, thus h ij L = L ij = L ij = h ij L. Lemma 2. If Lx, y) = Lx, y) + βx, y) is a projective Randers change, then 4) L G lijk + 2 L 2l lp ijk n + )L h ilg jk + h jl G ik + h kl G ij ) = LḠ lijk + 2 L2 l l Pijk Proof. From 3) one obtains hil Ḡ jk + n + ) L h jl Ḡ ik + h ) kl Ḡ ij. L h αldijk α = L g αl l α l l ) G α ijk n + )L G ijkh αl y α + h il G jk + h jl G ik + h kl G ij ). From the property h αl y α = 0 it follows that L h αldijk α = L g αl l α l l ) G α ijk n + )L h ilg jk + h jl G ik + h kl G ij ). From the definition of the hv-torsion see 2)) we conclude that L h αldijk α = g αl y ) α L L l l G α ijk n + )L h ilg jk + h jl G ik + h kl G ij ) = L G lijk + 2 L 2 l lp ijk n + )L h ilg jk + h jl G ik + h kl G ij ). The Douglas tensor D ijk is projective invariant. Moreover, by Lemma we have L h αldijk α = L hαl Dα ijk and this fact completes the proof. In the next two sections we give two consequences of the relation 4). 3. Projective Randers change between Landsberg spaces Definition. If a Finsler space satisfies the condition P ijk = 0, we call it a Landsberg space.

4 68 S. BÁCSÓ AND Z. KOVÁCS Theorem. Let F n and F n be Landsberg spaces and let Lx, y) = Lx, y) + βx, y) be a projective Randers change between them. Then where λx, y) is a scalar field. G lk Ḡlk = n h klλx, y). Proof. Let F n and F n be Landsberg spaces, i.e. P ijk = P ijk = 0. Then 4) becomes L G lijk n + )L h ilg jk + h jl G ik + h kl G ij ) = LḠ lijk hil Ḡ jk + n + ) L h jl Ḡ ik + h ) kl Ḡ ij. Moreover, for Landsberg spaces we have G lijk G iljk = 0, Ḡ lijk Ḡiljk = 0. These properties lead to n + )L h ijg lk + h ik G lj h jl G ik h kl G ij ) Hence we see that = ) hij Ḡ lk + h ik Ḡ lj h jl Ḡ ik h kl Ḡ ij. n + )L h ji Glk Ḡlk) + hki Glj Ḡlj) hjl Gik Ḡik) hkl Gij Ḡij) = 0. Contraction with g ij gives n ) G lk Ḡlk) + h α k Glα Ḡlα) ) h α l Gαk Ḡαk Denoting g ji G ij Ḡij) by λx, y) we find that h kl g ji G ij Ḡij) = 0. n ) G lk Ḡlk) + Glk Ḡlk) Glk Ḡlk) hkl λx, y) = Projective Randers change between weakly-berwald spaces Definition [2]). If a Finsler space satisfies the condition G ij = 0, we call it a weakly-berwald space. Theorem 2. Let F n and F n be two weakly Berwald Finsler spaces which are related by a projective Randers change L L. Let px, y) denote the projective factor of the change, that is Ḡi = G i +px, y)y i. Then i px, y) does not depend on y.

5 PROJECTIVE RANDERS CHANGES OF SPECIAL FINSLER SPACES 69 Proof. The equation 4) for a weakly-berwald space becomes: L G lijk + 2 L 2 l lp ijk = LḠ lijk + 2 L2 l l P ijk. Because of L G lijk + 2 L 2l lp ijk = L g αlg α ijk l l L 2 y αg α ijk = [ gαl l l l α )G α L ijk] we have L h lαg α ijk = L hlα Ḡ α ijk. Then it follows from h lα /L = h lα / L that that is L h lαg α ijk = L h lαḡα ijk, 5) 0 = h lα Ḡα ijk G α ijk). After successive derivations we have: Ḡ i = G i + px, y)y i Ḡ j i = Gi j + p j y i + pδ i j Ḡ i jk = G i jk + p jk y i + p j δ i k + p k δ i j Ḡ α ijk = G α ijk + p ijk y α + p jk δ α i + p ik δ α j + p ij δ α k. Substituting the last formula into 5) we have 0 = h lα pijk y α + p jk δi α + p ikδj α + p ijδk α ) 0 = h li p jk + h lj p ik + h lk p ij. By contracting with g li we obtain 0 = n )p jk + p jk + p jk. This shows that n + )p jk = 0 therefore p j x, y) does not depend on y. References [] S. Bácsó and I. Papp. P-Finsler spaces with vanishing Douglas tensor. Acta Acad. Paedagog. Agriensis, Sect. Mat. N.S.), 25:90 95, 998. [2] S. Bácsó and R. Yoshikawa. Weakly-Berwald spaces. Publ. Math. Debrecen, 6-2):29 23, [3] D. Bao, S.-S. Chern, and Z. Shen. An introduction to Riemann-Finsler geometry, volume 200 of Graduate Texts in Mathematics. Springer-Verlag, New York, [4] M. Hashiguchi and Y. Ichijyō. Randers spaces with rectilinear geodesics. Rep. Fac. Sci. Kagoshima Univ., 3):33 40, 980. [5] M. Matsumoto. On Finsler spaces with Randers metric and special forms of important tensors. J. Math. Kyoto Univ., 4: , 974. [6] M. Matsumoto. Finsler geometry in the 20th-century. In Handbook of Finsler geometry. Vol., 2, pages Kluwer Acad. Publ., Dordrecht, [7] H.-S. Park and I.-Y. Lee. The Randers changes of Finsler spaces with α, β)-metrics of Douglas type. J. Korean Math. Soc., 383):503 52, 200.

6 70 S. BÁCSÓ AND Z. KOVÁCS S. Bácsó University of Debrecen, 400 Debrecen, Pf. 2, Hungary address: Z. Kovács College of Nyíregyháza, 4400 Nyíregyháza, Sóstói út 3/b Hungary address: