Browsing by Author "Özdeğer, Abdülkadir"

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A Characterization of Totally Umbilical Hypersurfaces of a Space Form by Geodesic Mapping
(Springer, 2013) Canfes, Elif Ozkara; Özdeğer, Abdülkadir
The idea of considering the second fundamental form of a hypersurface as the first fundamental form of another hypersurface has found very useful applications in Riemannian and semi-Riemannian geometry especially when trying to characterize extrinsic hyperspheres and ovaloids. Recently T. Adachi and S. Maeda gave a characterization of totally umbilical hypersurfaces in a space form by circles. In our paper we give a characterization of totally umbilical hypersurfaces of a space form by means of geodesic mapping.
Citation - WoS: 0
Citation - Scopus: 0
Chebyshev Nets Formed by Ricci Curves in a 3-Dimensional Weyl Space
(Elsevier Science Bv, 2005) Yıldırım, Gülçin Çivi; Özdeğer, Abdülkadir
In this paper Ricci curves in a 3-dimensional Weyl space W-3(g T) are defined and it is shown that any 3-dimensional Chebyshev net formed by the three families of Ricci curves in a W-3(g T) having a definite metric and Ricci tensors is either a geodesic net or it consists of a geodesic subnet the members of which have vanishing second curvatures. In the case of in indefinite Ricci tensor only one of the members of the geodesic subnet under consideration has a vanishing second curvature. (c) 2004 Elsevier B.V. All rights reserved.
Citation - Scopus: 6
Conformal and Generalized Concircular Mappings of Einstein-Weyl Manifolds
(2010) Özdeğer, Abdülkadir
In this article after giving a necessary and sufficient condition for two Einstein-Weyl manifolds to be in conformal correspondence we prove that any conformal mapping between such manifolds is generalized concircular if and only if the covector field of the conformal mapping is locally a gradient. Using this fact we deduce that any conformal mapping between two isotropic Weyl manifolds is a generalized concircular mapping. Moreover it is shown that a generalized concircularly flat Weyl manifold is generalized concircular to an Einstein manifold and that its scalar curvature is prolonged covariant constant. © 2010 Wuhan Institute of Physics and Mathematics.
Citation - WoS: 3
Citation - Scopus: 5
Generalized Einstein Tensor for a Weyl Manifold and Its Applications
(Springer Heidelberg, 2013) Özdeğer, Abdülkadir
It is well known that the Einstein tensor G for a Riemannian manifold defined by R (alpha) (beta) = g (beta gamma) R (gamma I +/-) where R (gamma I +/-) and R are respectively the Ricci tensor and the scalar curvature of the manifold plays an important part in Einstein's theory of gravitation as well as in proving some theorems in Riemannian geometry. In this work we first obtain the generalized Einstein tensor for a Weyl manifold. Then after studying some properties of generalized Einstein tensor we prove that the conformal invariance of the generalized Einstein tensor implies the conformal invariance of the curvature tensor of the Weyl manifold and conversely. Moreover we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogonal trajectories of which are geodesics. Finally a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.
Citation - WoS: 7
Citation - Scopus: 8
On Sectional Curvatures of a Weyl Manifold
(2006) Özdeğer, Abdülkadir
In this paper it is proved that if at each point of a Weyl manifold the sectional curvature is independent of the plane chosen then the Weyl manifold is locally conformal to an Einstein manifold and that the scalar curvature of the Weyl manifold is prolonged covariant constant.