Chebyshev Nets Formed by Ricci Curves in a 3-dimensional Weyl Space
by
Gulcin Civi
Istanbul Technical University

Abstract

An n-dimensional differentiable manifold W_n is said to be a Weyl space if it has a conformal metric tensor g_ij and a symmetric connection Ñ_\gamma satisfying the compatibility condition given by the equation

Ñ\gamma g\alpha\beta-2 T\gamma g\alpha\beta=0 ,

(1.1)

where T_\gamma denotes a covariant vector field. Under the renormalization

~ g=\lambda2 g

(1.2)

of the metric tensor g, T is transformed by the law

~ T\gamma=T\gamma+\partial\gammaln\lambda

(1.3)

where \lambda is a function defined on W_n.

Let R_ij be the components of the Ricci tensor of the 3-dimensional Weyl space W₃(g, T) and let R_(ij) be the symmetric part of R_ij. Let the principal directions and the corresponding principal values of R_(ij) be denoted, respectively, by v[ || (1)] ,   v[ || (2)] ,   v[ || (3)]  and M[ || (1)] , M[ || (2)] , M[ || (3)] . Then, We get

(R(ij)+ M r   gij)  v r i=0  , (i, r=1, 2, 3)

We call v[ || (1)] ,   v[ || (2)]   and v[ || (3)]  the Ricci's principal directions and the integral curves of theese vector fields will be named as the Ricci curves of W₃(g, T). Theese curves may be considered as the generalization of Ricci curves in a Riemannian space.

In this paper, it is shown that any 3-dimensional Chebyshev net formed by the three families of Ricci curves in a W₃(g, T) having a definite metric and a Ricci tensor is either a geodesic net or it consists of a geodesic subnet the member of which have vanishing second curvatures. In the case of an indefinite Ricci tensor only one of the members of the geodesic subnet under consideration has a vanishing second curvature.

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Mathematics Subject Classification : Primary 53A40, Secondary : 53A30

Key Words : Weyl Space, Chebyshev Net , Ricci Curves.

Date received: July 17, 2002