Submanifolds of Lorentzian paracontact manifolds and their biharmonicities

This study which is designed as a philosophy doctoral thesis covers four chapter.In the first chapter we give some basic concepts such as harmonic and biharmonicmaps between Riemannian manifolds, nonexistence theorems for biharmonic submanifolds,biharmonic curves, semi-Riemannian manifolds, Lorentzian almost paracontactmanifolds and their submanifolds for the rest of the thesis that readers can easilyunderstand. The other chapters are the original parts of this thesis.The second chapter is devoted to the biharmonic curves in Lorentzian para-Sasakianmanifolds. In this chapter firstly by expressing the fact that n-dimensional(n ?4) conformal flat, quasi-conformal flat and conformal symmetric Lorentzianpara-Sasakian manifolds are locally isometric to n-dimensional Lorentzian unit sphere , we give Frenet formulas for spacelike and timelike curves in 4-dimensional Lorentzianpara-Sasakian manifolds. After then we obtain biharmonic equations for spacelike and timelike curves in 4-dimensional conformal flat, quasi-conformal flat and conformalsymmetric Lorentzian para-Sasakian manifolds. Moreover, by investigating thenecessary and sufficient conditions for spacelike and timelike curves in a 4-dimensional Lorentziansphere to be biharmonic, we examine the solutions of the obtained biharmonicequations in some special cases. So we show the existence of such curves.In the third chapter we study the invariant and non-invariant hypersurfacesof Lorentzian paracontact manifolds and biharmonic hypersurfaces of Lorentzianpara-Sasakian manifolds. We firstly investigate the non-invariant hypersurfaces ofalmost paracontact manifolds when the characteristic vector field of the manifolddoes not belong to the hypersurface and show that such hypersurfaces admit analmost product structure induced by the almost paracontact structure of the ambientmanifold. After then some characterizations on the invariant and non-invarianthypersurfaces of affinely cosymplectic and normal almost paracontact manifoldsare given. We prove that a non-invariant hypersurface of a Lorentzian almostparacontact manifold with the characteristic vector field nowhere tangent to thehypersurface is an almost product metric manifold. We also investigate the necessaryand sufficient conditions for a non-invariant hypersurface of a Lorentzian para-Sasa -kian manifold with the characteristic vector field nowhere tangent to the hypersurfaceto be locally product manifold. Moreover we give some examples for the hypersurfaceswhich are studied in this chapter and study the biharmonic spacelike and timelikehypersurfaces of Lorentzian para-Sasakian manifolds.In the fourth chapter we introduce the slant and semi-slant submanifolds ofLorentzian paracontact manifolds and give examples. In special we investigate theintegrability conditions for the distributions involved in the definition of a semi-slantsubmanifold when the ambient manifold is a Lorentzian paracosymplectic manifoldand a Lorentzian para-Sasakian manifold, respectively. We also study the warpedproduct, warped product semi-slant and warped product anti-slant submanifoldsof Lorentzian paracosymplectic manifolds and give some nonexistence theorems forsuch submanifolds in some special cases. In this chapter we finally investigate thebiharmonic submanifolds of Lorentzian para-Sasakian space forms.