الفهرس 
Di¤erentiable manifoldsOnly 14 pages are availabe for public view
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Abstract
The group of all smooth transformations of a Riemannian manifold is a very important group. This group leads us to study a subset of these transformations which preserve a certain geometric or physical quantity, say , invariant. The flow of each such transformation generates a vector field say𝜉. In this case the Lie derivative of 𝐺 vanishes in direction of 𝜉 i.e.𝐿ξ𝐺 = 0.This thesis is devoted to the study this class of vector fields and their generalizations. The study of conformal vector fields and their generalizations represents a fruitful research area. Consequently, a growing body of research is concerned with their properties and the geometric implications for their existence on Riemannian manifolds. This thesis is concerned with the study of vector fields and their geometrical characteristics on(semi)Riemannian product manifolds and (semi)Riemannian warped product manifolds. The study also considered providing different generalizations of these vector fields to the(semi)Riemannian warped product manifolds. Chapter one deals with the presentation of the basic geometrical concepts on(semi)Riemannian manifolds. For example, we presented the concepts of vector fields,connections, curvature and Ricci curvature. Also, the concept of Lie derivatives of functions and tensors is discussed. This chapter presents also the definitions and examples of both Killing vector fields and conformal vector fields in (semi-)Riemannian manifolds. Many geometric characteristics and examples of these aforementioned vector fields and many geometrical effects of their presence on (semi-)Riemannian manifolds are considered. A generalized Robertson-Walker spacetime is usually pictured out as a special case of warped product manifolds. Chapter two is concerned with the study of symmetries of generalized Robertson-Walker spacetimes generated by conformal vector fields and different collineations. Conditions on the warped product factor are derived to guarantee the existence of conformal vector fields and collineations on generalized Robertson Walker spacetimes. Finally, Ricci soliton structure on generalized Robertson-Walker spacetimes generated by conformal vector fields is considered.