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 ....

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 D

L

JK

L = 0, and a *PRanders space with a closed one-form ß(x,y) is a Finsler space with vanishing Douglas tensor Theorem 6. ([3]) A Randers space is a Douglas space iff ß(x,y) is a closed form. Then /i-I \ r\/~i% i 7 k I rlrn.y y i (n) 26 = i

3kV

Jy + a + ß y > where 7j

k(x) is the Levi-Civita connection of a Riemannian space, ri

m is equal to b

i;

j hence ri

m depends only on position. From the Theorem 6. and (10) follows that *Wy

m =

ey(r)

(Q + /3

) a + ß that is L From the last equation we obtain rimy

ly

m = e^L

2. DijfFerentiating twice this equation by y

l and y

m we get b

vj = e^g U' This means that the metrical tensor g

t

J 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 *PKropina 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.