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Akyol [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistics 2017; 462: 177-192] defined and studied conformal antiinvariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant submersions (it means the Reeb vector field $\xi$ is a vertical vector field) from cosymplectic manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions, and conformal antiinvariant submersions. More precisely, we mention many examples and obtain the geometries of the leaves of vertical distribution and horizontal distribution, including the integrability of the distributions, the geometry of foliations, some conditions related to total geodesicness, and harmonicity of the submersions. Finally, we consider a decomposition theorem on the total space of the new submersion.

In this paper, we introduce the concept of framed-complex submersion from a framed metric manifold onto an almost Hermitian manifold.We investigate the influence of a given structure defined on the total manifold on the determination of the corresponding structure on the base manifold. Moreover, we provide an example, investigate various properties of theO’Neill’s tensors for such submersions, find the integrability of the horizontaldistribution. We also obtain curvature relations between the base manifoldand the total manifold

Bu tez dört bölümden meydana gelmektedir. Birinci bölümde konunun tarihsel gelişimi ve ele alınan problemlerin tanıtımı yapılmaktadır. İkinci bölümde diğer bölümlere faydalı olacak temel tanım ve kavramlar; (semi-Riemann manifoldlar, vektör demetleri ve distribüsyonlar, semi-Riemannian submersiyonlar, çarpım Riemann manifoldu, hemen hemen para-kontak metrik manifoldlar) ele alınmaktadır.Üçüncü bölümde, hemen hemen çarpım submersiyonlar çalışılmaktadır. Hemen hemen Riemann çarpım manifoldları arasında hemen hemen çarpım Riemann submersiyon tanımlanmakta ve bir örnek verilmektedir. Daha sonra bu submersiyonlar için O'Neill tensörleri tanımlanmakta ve bunların temel özellikleri incelenmektedir. Ayrıca, bir hemen hemen çarpım Riemann submersiyonun total uzay, baz uzay ve liflerinin bi-kesit ve kesit eğrilikleri arasındaki ilişki araştırılmaktadır.Dördüncü bölümde, para-kontakt semi-Riemann submersiyonlara ayrılmaktadır. Hemen hemen para-kontakt metrik manifoldları arasında para-kontakt semi-Riemann submersiyon tanımlanmakta ve bir örnek verilmektedir. Ayrıca bu submersiyonlar için O'Neill tensörleri tanımlanmakta ve bunların temel özellikleri incelenmektedir. Daha sonra yatay distribüsyonun integrallenebilir ve liflerin total jeodezikliği için gerek şart elde edilmektedir. Total manifoldun sahip olduğu para-kontakt yapınınn para-kontakt submersiyon tarafından baz manifolda taşınması durumu araştırılmaktadır. Son olarak, bir para-kontakt semi-Riemann submersiyonun total uzay, baz uzay ve liflerinin bi-kesit ve kesit eğrilikleri incelenmektedir. Total manifoldun bazı özel durumları için eğrilikler arasındaki bağıntıların oldukça basit bir duruma indirgendiği elde edilmektedir. 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. 96

In this paper, we introduce the idea of a lightlike submersion from a semi-Riemannian manifold onto a lightlike manifold, and give some examples. Then we define O’Neill’s tensors for such submersions and investigate their main properties. We show that the Schouten connection is not a metric connection in a lightlike submersion. We also investigate curvature properties of the manifolds and establish a relation between the null sectional curvatures of a semi-Riemannian manifold and a lightlike manifold.

The first chapter of this thesis is devoted some basic topics, such as, Riemannianmanifold, connection, curvatures, vector bundles and distributions. These terms are necessaryfor the other chapters of this thesis.In the second chapter, after the basic terminology, the fundamental tensor fields for aRiemannian submersion are given and their basic properties are studied. Afterwardsgeometric meaning of basic tensors are investigated.In the last chapter, firstly the covariant derivative of basic tensors are found andtheir basic properties are studied. Then the relation between two curvatures of Riemannmanifolds are found by using these properties.KEYWORDS: Riemann submersion, basic tensor, covariant derivative, curvature, Riccitensor, scalar curvature, distribution, vector bundle. 95