Lorentzian hemen hemen parakontakt manifoldların altmanifoldları ve biharmoniklikleri
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Dosyalar
Tarih
2011
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Yayıncı
İnönü Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Doktora tezi olarak hazırlanan bu çalışma dört bölümden oluşmaktadır. Birinci bölüm diğer bölümlerin daha iyi anlaşılabilmesi için bazı temel kavramlara ayrıldı. Diğer bölümler ise tezin orjinal kısımlarıdır. Birinci bölümde manifoldlar arasındaki harmonik ve biharmonik dönüşümler, biharmonik altmanifoldlar, biharmonik eğriler, semi-Riemann manifoldlar, Lorentzian hemen hemen parakontakt manifoldlar ve altmanifoldlar ile ilgili temel tanım ve teoremler ele alındı. İkinci bölüm Lorentzian para-Sasakian manifoldlar üzerindeki biharmonik eğriler çalışıldı. Bu bölümde öncelikle n-boyutlu(n ?4) konformal flat, kuasi-konformal flat ve konformal simetrik Lorentzian para-Sasakian manifoldlarınn-boyutlu birim Lorentzian küresine lokal olarak izometrik olduğu ifade edilerek 4-boyutlu Lorentzian para-Sasakian manifoldlar üzerindeki spacelike ve timelike eğriler için Frenet formülleri verildi. Daha sonra 4-boyutlu konformal flat, kuasi-konformal flatve konformal simetrik Lorentzian para-Sasakian manifoldların sırasıyla, spacelike vetimelike eğrileri için biharmonik denklemler elde edildi. Son olarak 4-boyutlu Lorentzian birim küresi üzerindeki spacelike ve timelike eğrilerin biharmonik olması için gerek ve yeter şartlar incelenerek elde edilen biharmonik denklemlerin bazı özel durumlariçin çözümleri irdelendi ve bu tip eğrilerin varlığı araştırıldı. Üçüncü bölümde Lorentzian hemen hemen parakontakt manifoldların invaryant, non-invaryant hiperyüzeyleri ve Lorentzian para-Sasakian manifoldların biharmonik hiperyüzeyleri incelendi. Bu bölümde ilk olarak hemen hemen parakontakt manifoldların, karakteristik vektör alanının hiperyüzeye ait olmaması durumunda non-invaryant hiperyüzeyleri ele alınarak bu tip hiperyüzeylerin hemen hemen parakontakt yapıdan indirgenen bir hemen hemen çarpım yapısına sahip olduğu gösterildi. Daha sonra afin kosimplektik ve normal hemen hemen parakontakt manifoldların invaryant ve non-invaryant hiperyüzeyleri için bazı karakterizasyonlar verildi. Ayrıca Lorentzian hemen hemen parakontakt manifoldların, karakteristik vektör alanının hiperyüzeye ait olmaması durumunda non-invaryant hiperyüzeylerinin bir hemen hemen çarpım metrik manifoldu olduğu gösterildi ve Lorentzian para-Sasakian manifoldların, bu tip hiperyüzeylerinin ise bir lokal çarpım manifoldu olması için gerek ve yeter şartlar elde edildi. Bu bölümde incelenen hiperyüzeylere örnekler verildikten sonra Lorentz para-Sasakian manifoldların spacelike ve timelike hiperyüzeylerinin biharmonik olması için gerek ve yeter şartlar araştırıldı. Dördüncü bölümde Lorentzian hemen hemen parakontakt manifoldların slant ve semi-slant altmanifoldları tanıtılarak bu altmanifoldlara örnekler verildi. Özel olarak manifoldun Lorentzian parakosimplektik ve Lorentzian para-Sasakian olması durumunda semi-slant altmanifoldların tanımında yer alan distribüyonların integrallenebilirlik şartları incelendi. Ayrıca Lorentzian parakosimplektik manifoldların warped çarpım, warped çarpım semi-slant ve warped çarpım anti-slant altmanifoldları ele alınarak bazı özel durumlarda bu altmanifoldların yokluğu ile ilgili sonuçlar elde edildi. Bu bölümde son olarak Lorentzian para-Sasakian uzay formların biharmonik altmanifoldları incelendi.
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 biharmonic maps between Riemannian manifolds, nonexistence theorems for biharmonic submanifolds, biharmonic curves, semi-Riemannian manifolds, Lorentzian almost paracontact manifolds and their submanifolds for the rest of the thesis that readers can easily understand. The other chapters are the original parts of this thesis. The second chapter is devoted to the biharmonic curves in Lorentzian para-Sasakian manifolds. In this chapter firstly by expressing the fact that n-dimensional (n ?4) conformal flat, quasi-conformal flat and conformal symmetric Lorentzian para-Sasakian manifolds are locally isometric to n-dimensional Lorentzian unit sphere , we give Frenet formulas for spacelike and timelike curves in 4-dimensional Lorentzian para-Sasakian manifolds. After then we obtain biharmonic equations for spacelike and timelike curves in 4-dimensional conformal flat, quasi-conformal flat and conformal symmetric Lorentzian para-Sasakian manifolds. Moreover, by investigating the necessary and sufficient conditions for spacelike and timelike curves in a 4-dimensional Lorentzian sphere to be biharmonic, we examine the solutions of the obtained biharmonic equations in some special cases. So we show the existence of such curves. In the third chapter we study the invariant and non-invariant hypersurfaces of Lorentzian paracontact manifolds and biharmonic hypersurfaces of Lorentzian para-Sasakian manifolds. We firstly investigate the non-invariant hypersurfaces of almost paracontact manifolds when the characteristic vector field of the manifold does not belong to the hypersurface and show that such hypersurfaces admit an almost product structure induced by the almost paracontact structure of the ambient manifold. After then some characterizations on the invariant and non-invariant hypersurfaces of affinely cosymplectic and normal almost paracontact manifolds are given. We prove that a non-invariant hypersurface of a Lorentzian almost paracontact manifold with the characteristic vector field nowhere tangent to the hypersurface is an almost product metric manifold. We also investigate the necessary and sufficient conditions for a non-invariant hypersurface of a Lorentzian para-Sasa - kian manifold with the characteristic vector field nowhere tangent to the hypersurface to be locally product manifold. Moreover we give some examples for the hypersurfaces which are studied in this chapter and study the biharmonic spacelike and timelike hypersurfaces of Lorentzian para-Sasakian manifolds. In the fourth chapter we introduce the slant and semi-slant submanifolds of Lorentzian paracontact manifolds and give examples. In special we investigate the integrability conditions for the distributions involved in the definition of a semi-slant submanifold when the ambient manifold is a Lorentzian paracosymplectic manifold and a Lorentzian para-Sasakian manifold, respectively. We also study the warped product, warped product semi-slant and warped product anti-slant submanifolds of Lorentzian paracosymplectic manifolds and give some nonexistence theorems for such submanifolds in some special cases. In this chapter we finally investigate the biharmonic submanifolds of Lorentzian para-Sasakian space forms.
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 biharmonic maps between Riemannian manifolds, nonexistence theorems for biharmonic submanifolds, biharmonic curves, semi-Riemannian manifolds, Lorentzian almost paracontact manifolds and their submanifolds for the rest of the thesis that readers can easily understand. The other chapters are the original parts of this thesis. The second chapter is devoted to the biharmonic curves in Lorentzian para-Sasakian manifolds. In this chapter firstly by expressing the fact that n-dimensional (n ?4) conformal flat, quasi-conformal flat and conformal symmetric Lorentzian para-Sasakian manifolds are locally isometric to n-dimensional Lorentzian unit sphere , we give Frenet formulas for spacelike and timelike curves in 4-dimensional Lorentzian para-Sasakian manifolds. After then we obtain biharmonic equations for spacelike and timelike curves in 4-dimensional conformal flat, quasi-conformal flat and conformal symmetric Lorentzian para-Sasakian manifolds. Moreover, by investigating the necessary and sufficient conditions for spacelike and timelike curves in a 4-dimensional Lorentzian sphere to be biharmonic, we examine the solutions of the obtained biharmonic equations in some special cases. So we show the existence of such curves. In the third chapter we study the invariant and non-invariant hypersurfaces of Lorentzian paracontact manifolds and biharmonic hypersurfaces of Lorentzian para-Sasakian manifolds. We firstly investigate the non-invariant hypersurfaces of almost paracontact manifolds when the characteristic vector field of the manifold does not belong to the hypersurface and show that such hypersurfaces admit an almost product structure induced by the almost paracontact structure of the ambient manifold. After then some characterizations on the invariant and non-invariant hypersurfaces of affinely cosymplectic and normal almost paracontact manifolds are given. We prove that a non-invariant hypersurface of a Lorentzian almost paracontact manifold with the characteristic vector field nowhere tangent to the hypersurface is an almost product metric manifold. We also investigate the necessary and sufficient conditions for a non-invariant hypersurface of a Lorentzian para-Sasa - kian manifold with the characteristic vector field nowhere tangent to the hypersurface to be locally product manifold. Moreover we give some examples for the hypersurfaces which are studied in this chapter and study the biharmonic spacelike and timelike hypersurfaces of Lorentzian para-Sasakian manifolds. In the fourth chapter we introduce the slant and semi-slant submanifolds of Lorentzian paracontact manifolds and give examples. In special we investigate the integrability conditions for the distributions involved in the definition of a semi-slant submanifold when the ambient manifold is a Lorentzian paracosymplectic manifold and a Lorentzian para-Sasakian manifold, respectively. We also study the warped product, warped product semi-slant and warped product anti-slant submanifolds of Lorentzian paracosymplectic manifolds and give some nonexistence theorems for such submanifolds in some special cases. In this chapter we finally investigate the biharmonic submanifolds of Lorentzian para-Sasakian space forms.
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Yüksel Perktaş, S. (2011). Lorentzian hemen hemen parakontakt manifoldların altmanifoldları ve biharmoniklikleri. İnönü Üniversitesi Fen Bilimleri Enstitüsü. 1-193 ss.
