ON THE GEOMETRY OF WARPED PRODUCT SPACE-TIMES

Abstract

The concept of smooth manifolds is a fruitful generalization of the surfaces in the Euclidean space. This concept is widely used in general relativity to generate different space-times. A space-time is pictured out as a 4-dimensional Lorentzian smooth manifold. Thus, the study of the geometry of smooth manifolds as well as space-times is of particular interest and has become a favorite topic for mathematicians and physicists. In this paper, we give some of the necessary background in differential geometry, relativity, and cosmology. This background is essential in the whole thesis. First, the concepts of differentiable as well as Riemannian manifolds are investigated. Then, we introduce the concept of warped product manifolds. Finally, as a special case of warped product manifolds, we study the geometry of generalized Robertson-Walker space-times. Einstein manifolds and their generalizations are considered. Also, Einstein-like manifolds are investigated.