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

92 Sándor Bácsó and Ildikó Papp 2. Douglas tensor, Randers metric, *P-space Let us consider two Finsler space F

n (M

n,Z) and F L (M

n, L) on a ——71 common underlying manifold M

n. A diffeomorphism F

n -> F is called 71 geodesic if it maps an arbitrary geodesic of F

n to a geodesic of F .In this case the change L —» L of the metric is called projective. It is well-known that the mapping F

n —F is geodesic iff there exist a scalar field p(x,y) satisfying the following equation (2) G

2 = G

l + 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 + pj6

l

k + p

kb) -f p

j

ky\ Pj

k = p

m, (5) G)

k

l = G)

k

l + Pjktf + Vji^k + PkiSj + Pjkiy

1, Pjki = Pjk(i)Substituting pij = (Gij - Gij) /(n + 1) and p

lj

k = (Gij(

k) - G

ij{

k)) /(n + 1) into (5) we obtain the so called Douglas tensor which is invariant under geodesic mappings, that is (6) D)

k

i = G)

k

l - (y'Gjkd) + 6

ljG

k

l + S

l

kGji + 6}G

j

k) /(n + 1), which is invariant under geodesic mappings, that is (7) D)

k

l = D)

k

l. We now consider some notions and theorems for special Finsler spaces. Definition 1. ([1]) In an n-dimensional differentiable manifold M

n a Finsler metric L(x,y) = a(x,y) + ß(x,y) is called Randers metric, where a(x, y) = y/cLij{x)y

ly

J is a Riemannian metric in M

n and ß(x, y) = b

l(x)y

l is a differential 1-form in M

n. The Finsler space F

n = (M

n, L) = a + ß with Randers metric is called Randers space. Definition 2. ([1]) The Finsler metric L — a

2 /

ß is called Kropina metric. The Finsler space F

n = (M

n,X) = a

21

ß with Kropina metric is called Kropina space.