On the geometry of product submersions

This thesis consists of four chapters. In the first chapter the motivation of the problem is presented. In the second chapter, we give basic materials such as semi-Riemannian manifolds, distributions, vector bundles, (semi)-Riemannian submersion, almost product manifolds and almost para-contact metric manifolds which will be useful for other chapters.In the third chapter, we first define the concept of almost product submersion between almost product manifolds, then we provide an example and show that the vertical and horizontal distributions of such submersions are invariant with respect to the almost product structure of the total manifold. We also prove that if the total manifold of the almost product submersion is a locally product Riemannian manifold, then the base manifold is also a locally product Riemannian manifold. Moreover, we obtain various properties of the O'Neill's tensors for such submersions and find the integrability of the horizontal distribution. Finally, we obtain curvature relations between the base manifold and the total manifold.In the fourth chapter, we first define the concept of paracontact semi-Riemannian submersions between almost paracontact metric manifolds, then we provide an example and show that the vertical and horizontal distributions of such submersions are invariant with respect to the almost paracontact structure of the total manifold. The study is focused on fundamental properties and the transference of structures defined on the total manifold. Moreover, we obtain various properties of the O'Neill's tensors for such submersions and find the integrability of the horizontal distribution. We also find necessary and sufficient conditions for a paracontact semi-Riemannian submersion to be totally geodesic.Finally, we obtain curvature relations between the base manifold and the totalmanifold.