Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)
BACSÓ , S . and PAPP, L, P-Finsler spaces with vanishing Douglas tensor ....
*P-Finsler spaces with vanishing Douglas tensor 93 Definition 3. ([1], [6]) A Finsler space of dimension n > 2 is called C-reducible, if the tensor Cijk = \9ij(k) ca n be written in the form (8) Cijk — —~ 7 (h-ijCk + h lkC J + hjkCi) , 71+1 where hij = g Z ] — IJj is the angular metric tensor and = Luy Theorem 1. ([7]) A Finsler space F n , n > 3, is C-reducible iff the metric is a Randers metric or a Kropina metric. Definition 4. ([4], [5]) A Finsler space F n is called *P-Finsler space, if the tensor P l 3k — \9ij-,k can be written in the form (9) Pijk = \{x,y)C lj k. 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 Definition 5. ([3]) A Finsler space is said to be of Douglas type or Douglas space, iff the functions G ly 3 - G ]y x are homogeneous polynomials in (y l) of degree three. Theorem 3. ([3]) A Finsler space is of Douglas type iff the Douglas tensor vanishes identically. Theorem 4. ([5]) For n > 3, in a C-reducible *P-Finsler space D l jk l — 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. Definition 6. ([1]) A Finsler space is called Landsberg space if the condition P Z Jk — 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 (fulfilling the condition P l] k — p(x,y)C l Jk), then p(x,y) can be given by the equation (10) p(x,y) = e^L(x,y).
