Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

GÁT, G., On a theorem of type Hardy-Littlewood with respect to the Vilenkin-like systems

*P-Finsler spaces with vanishing Douglas tensor S. BÁCSÓ, I. PAPP Abstract. 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. 1. Introduction Let F

n (M

n

, L

) be an n-dimensional Finsler space, where M

n is a connected 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 x

1 = x^t),

1 (t

0 < t < t

x). The system of differential equations for geodesic curves of F

n with respect to canonical parameter t is given by d

2x

l „ „• dx

l dt> y d

t where 1 . ( fftq G' = 4®'

r "' hi r) 9ij = l

Lh)U)> (i) = (<7

U) = (9ij) 1 The Berwald connection coefficients G^x^y), G

l

]

k(x,y) can be derived from the function G\ namely G

* = G

1^ and G

l

j

k — G)(

ky The Berwald covariant derivative with respect to the Berwald connection can be written (1) Tj.

k = dr;/dx

k - Tj

(r

)G

r

k + T]G\

k - T

l

rG

r

j

k. (Throughout the present paper we shall use the terminology and definitions described in Matsumoto's monograph [6].) This work was partially supported by the Ministry of Culture and Education of Hungary under Grant No. FKFP 0457. 1 The Roman indices run over the range l,...,n.