The purpose of the present paper is to prove that a 'P-Randers space with vanishing Douglas tensor is a Riemannian space if the dimension is greater then three

Bácsó, Sándor

Papp, Ildikó

English

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*P-Finsler spaces with vanishing Douglas tensor S. BÁCSÓ, I. PAPP A b s t r a c t . The purpose of the present paper is to prove t h a t a ' P - R a n d e r s space with vanishing Douglas tensor is a Riemannian space if the dimension is g rea te r then three. 1. Introduction Let Fn (Mn, L) be an n-dimensional Finsler space, where Mn is a con-nected differentiable manifold of dimension n and y) is the fundamental function defined on the manifold T(M)\ 0 of nonzero tangent vectors. Let us consider a geodesic curve x1 = x^t),1 (t0 < t < tx). The system of differen-tial equations for geodesic curves of Fn with respect to canonical parameter t is given by d2xl „ „• dxl dt-> y d t where 1 . ( f f t q G' = 4®'r " ' hi r) 9ij = lLh)U)> (i) = (<7U) = (9ij) 1 The Berwald connection coefficients G^x^y), Gl]k(x,y) can be derived from the function G\ namely G* = G1^ and Gljk — G)(ky The Berwald covariant derivative with respect to the Berwald connection can be written (1) Tj.k = dr;/dxk - Tj(r)Grk + T]G\k - TlrGrjk. (Throughout the present paper we shall use the terminology and defi-nitions described in Matsumoto's monograph [6].) This work was partially suppor ted by the Ministry of Cu l tu re and E d u c a t i o n of Hungary under Grant No. F K F P 0457. 1 T h e Roman indices run over the range l , . . . ,n . 92 Sándor Bácsó and Ildikó Papp 2. Douglas tensor, Randers metric, *P-space Let us consider two Finsler space Fn ( M n , Z ) and F L ( M n , L) on a ——71 common underlying manifold Mn. A diffeomorphism Fn -> F is called 71 geodesic if it maps an arbitrary geodesic of Fn to a geodesic of F . I n this case the change L —» L of the metric is called projective. It is well-known that the mapping Fn —F is geodesic iff there exist a scalar field p(x,y) satisfying the following equation (2) G2 = Gl + p(x, y)y\ p ± 0. The projective factor p(x,y) is a positive homogeneous function of degree one in y. From (2) we obtain the following equations (3) G* = G) + pS) + pjy\ pj = p{j), (4) G)k = G)k + pj6lk + pkb) -f pjky\ Pjk = pm, (5) G)kl = G)kl + Pjktf + Vji^k + PkiSj + Pjkiy1, Pjki = Pjk(i)-Substituting pij = (Gij - Gij) / (n + 1) and pljk = (Gij(k) - Gij{k)) / (n + 1) into (5) we obtain the so called Douglas tensor which is invariant under geodesic mappings, that is (6) D)ki = G)kl - (y'Gjkd) + 6ljGkl + SlkGji + 6}Gjk) /(n + 1), which is invariant under geodesic mappings, that is (7) D)kl = D)kl. We now consider some notions and theorems for special Finsler spaces. Definit ion 1. ([1]) In an n-dimensional differentiable manifold Mn a Finsler metric L(x,y) = a(x,y) + ß(x,y) is called Randers metric, where a(x, y) = y/cLij{x)ylyJ is a Riemannian metric in Mn and ß(x, y) = bl(x)yl is a differential 1-form in Mn. The Finsler space Fn = ( M n , L) = a + ß with Randers metric is called Randers space. Definit ion 2. ([1]) The Finsler metric L — a2 / ß is called Kropina metric. The Finsler space Fn = (M n ,X) = a21ß with Kropina metric is called Kropina space. *P-Finsler spaces with vanishing Douglas tensor 9 3 Definit ion 3. ([1], [6]) A Finsler space of dimension n > 2 is called C-reducible, if the tensor Cijk = \9ij(k) c a n be written in the form (8) Cijk — — ~ 7 (h-ijCk + h lkCJ + hjkCi) , 7 1 + 1 where hij = gZ] — IJj is the angular metric tensor and = Luy Theorem 1. ([7]) A Finsler space Fn, n > 3, is C-reducible i f f the metric is a Randers metric or a Kropina metric. Definit ion 4. ([4], [5]) A Finsler space Fn is called *P-Finsler space, if the tensor Pl3k — \9ij-,k can be written in the form (9) Pijk = \{x,y)Cljk. Theorem 2. ([4]) For n > 3 in aC-reducible *P-Finsler space A(x, y) = k(x)L(x, y) holds and k(x) is only the function of position. 3. ^P-Randers space with vanishing Douglas tensor Definit ion 5. ([3]) A Finsler space is said to be of Douglas type or Douglas space, iff the functions Gly3 - G]yx are homogeneous polynomials in (yl) of degree three. Theorem 3. ([3]) A Finsler space is of Douglas type i f f the Douglas tensor vanishes identically. Theorem 4. ([5]) For n > 3, in a C-reducible *P-Finsler space Dljkl — 0 holds. If we consider a Randers change L(x,y) L(x,y) + ß{x,y), where ß(x ,y) is a closed one-form, then this change L —> L is projective. Definit ion 6. ([1]) A Finsler space is called Landsberg space if the condition PZJk — 0 holds. Theorem 5. ([2]) If there exist a Äanders change with respect to a projective scalar p(x.y) between a Landsberg and a *P-Finsler space (ful-filling the condition Pl]k — p(x,y)ClJk), then p(x,y) can be given by the equation (10) p(x,y) = e^L(x,y). 94 Sándor Bácsó and Ildikó Papp It is well-known that the Riemannian space is a special case of the Landsberg space. In a Riemannian space we have DLJKL = 0, and a *P-Randers space with a closed one-form ß(x,y) is a Finsler space with van-ishing Douglas tensor Theorem 6. ([3]) A Randers space is a Douglas space i f f ß(x,y) is a closed form. Then /i-I \ r\/~i% i 7 k I rlrn.y y i (n) 26 = i3kVJy + a + ß y > where 7 j k ( x ) is the Levi-Civita connection of a Riemannian space, rim is equal to bi;j hence rim depends only on position. From the Theorem 6. and (10) follows that * W y m = e y ( r ) ( Q + /3) a + ß that is L From the last equation we obtain rimylym = e^L2. DijfFerentiating twice this equation by yl and ym we get bvj = e ^ g U' This means that the metrical tensor gtJ depends only on x, so we get the following Theorem. A *P-RcLnders space with vanishing Douglas tensor is a Riemannian space if the dimension is greater than three. 4. Further possibilities From Theorem 1, Theorem 4 and our Theorem follows that only the *P-Kropina spaces can be *P-C reducible spaces with vanishing Douglas tensor which are different from Riemannian spaces. We would like to investigate this letter case in a forthcoming paper. P-Finsler spaces with vanishing Douglas tensor References 9 5 [1] P . L . A N T O N E L L I , R . S . I N G A R D E N , M . M A T S U M O T O , The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Acad. Publ . , Dor-drecht , Boston, London, 1993. [2] S. BÁCSÓ, On geodesic mapp ing of special Finsler spaces, Rendiconti Palermo ( to appear) . [3] S. BÁcsó , M. MATSUMOTO, On Finsler spaces of Douglas type, A generalisation of the notion of Berwald space. Publ. Math. Debrecen, 51 (1997), 385-406. [4] H. IZUMI, On * P-Finsler spaces I., II. Memoirs of the Defense Academy, Japan, 16 (1976), 133-138, 17 (1977), 1 -9 . [5] H. IZUMI, On *P-Fins ler spaces of scalar curvature , Tensor, N. S. 3 8 (1982), 220 222 . [6] M. MATSUMOTO, S. HOJO, A conclusive theorem on C-reduc ib le Finsler spaces, Tensor, N. S. 32 (1978), 225-230. S Á N D O R B Á C S Ó L A J O S K O S S U T H U N I V E R S I T Y I N S T I T U T E OF M A T H E M A T I C S AND I N F O R M A T I C S 4 0 1 0 D E B R E C E N P . O . B o x 1 2 H U N G A R Y E-mail: sbacsoimath.klte.hu I L D I K Ó P A P P L A J O S K O S S U T H U N I V E R S I T Y I N S T I T U T E OF M A T H E M A T I C S AND I N F O R M A T I C S 4 0 1 0 D E B R E C E N P . O . B o x 1 2 H U N G A R Y E-mail: ipapp8math.klte.hu 

*P-Finsler spaces with vanishing Douglas tensor.

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*P-Finsler spaces with vanishing Douglas tensor.

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