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Geometric Topology II
September 29 - October 5, 2002
Inter-University Center, Dubrovnik; Department of Mathematics, University of Zagreb
Dubrovnik, Croatia

Organizers
Ivan Ivansic, University of Zagreb;, James E. Keesling, University of Florida;, Alexander N. Dranishnikov, University of Florida;, Sime Ungar, University of Zagreb

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

Abstract




CHEBYSHEV NETS FORMED BY RICCI CURVES
IN A 3-DIMENSIONAL WEYL SPACE

Gülçin Çivi Yildirim


An n-dimensional differentiable manifold Wn is said to be a Weyl space if it has a conformal metric tensor gij 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 Wn.

Let Rij 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 Rij. 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

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caje-28.