Bazı Özel Manifoldlar Üzerinde Vektör Alanları
Bazı Özel Manifoldlar Üzerinde Vektör Alanları
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Tarih
2016-06-20
Yazarlar
Kırık, Bahar
Süreli Yayın başlığı
Süreli Yayın ISSN
Cilt Başlığı
Yayınevi
Fen Bilimleri Enstitüsü
Institute of Science And Technology
Institute of Science And Technology
Özet
Bu çalışmanın amacı, geometrik ve fiziksel açıdan önem taşıyan bazı manifoldlar üzerinde tanımlanabilen vektör alanlarını incelemek ve özelliklerini araştırmaktır. Çalışmanın birinci bölümünde, çalışmada göz önüne alınacak olan bazı manifoldlar ve özel vektör alanlarından bahsedilmiştir. Bu amaçla, ilk olarak, özel manifoldlar ile ilgili literatür taraması verilmiştir. Ayrıca, bu manifoldların ve vektör alanlarının, başta geometri ve fizik olmak üzere, diğer birçok bilim ve mühendislik dalındaki uygulama alanları vurgulanmıştır. Çalışmanın ikinci bölümünde, çalışmada kullanılacak olan bazı temel kavramlardan bahsedilmiştir. Daha sonra, ele alınan manifoldlar üzerinde incelenecek olan özel vektör alanlarının tanımlarına yer verilmiştir. Çalışmanın üçüncü bölümünde, genelleştirilmiş Einstein manifoldlarının çeşitleri incelenmiştir. İlk olarak, fiziksel açıdan önem taşıyan yarı-Einstein manifoldları üzerindeki özel vektör alanlarını inceleme problemi ele alınmıştır. Bununla birlikte, 4-boyutlu yarı-Einstein manifoldu örneği verilmiştir. Ayrıca, yarı-Einstein manifoldlarının konformal dönüşümleri araştırılmıştır. Bu konformal dönüşüm altında, $\varphi(\rm Ric)$ ve konsörkılır vektör alanlarının birtakım özellikleri elde edilmiş ve konformal dönüşümün aynı zamanda konharmonik olması halinde yarı-Einstein manifoldunun ilişkili büyüklükleri arasındaki bağıntılar bulunmuştur. Daha sonra, bu bölümde, neredeyse yarı-Einstein manifoldları göz önüne alınmıştır. Neredeyse yarı-Einstein manifoldlarının çeşitli geometrik özellikleri incelenmiştir. Bu manifoldun üreteçlerinin özel vektör alanları olabilme şartları araştırılmış ve bu manifoldlarla ilgili çeşitli teoremler ispatlanmıştır. Bunlara ek olarak, özel vektör alanları içeren neredeyse yarı-Einstein manifoldlarının konformal dönüşümleri göz önüne alınmıştır. Konformal ve konharmonik dönüşüm altında, konsörkılır ve $\varphi(\rm Ric)$ vektör alanları üzerine çeşitli sonuçlar elde edilmiştir. Ayrıca, bu manifoldun 4-boyutlu uzaylardaki örnekleri verilmiş ve neredeyse yarı-Einstein uzay-zamanı incelenmiştir. Bu bölümde son olarak, genelleştirilmiş yarı-Einstein manifoldları üzerindeki özel vektör alanlarını inceleme problemi ele alınmıştır. Bu manifoldlar üzerinde tors oluşturan vektör alanları ve $\varphi(\rm Ric)$ vektör alanları incelenmiştir. Manifoldun üreteç vektör alanlarının, söz konusu özel vektör alanları olup olmaması durumu araştırılmıştır. Bununla birlikte, Ricci tensörü için göz önüne alınan bazı özel koşullar altında, manifold üzerinde tanımlanan özel vektör alanlarının özellikleri incelenmiştir. Çalışmanın dördüncü bölümünde, pseudo Ricci simetrik ve hemen hemen pseudo Ricci simetrik manifoldlar göz önüne alınmıştır. Pseudo Ricci simetrik ve hemen hemen pseudo Ricci simetrik genelleştirilmiş yarı-Einstein manifoldların özellikleri araştırılmıştır. Pseudo Ricci simetrik manifoldlara ait bazı sonuçlardan faydalanılarak, genelleştirilmiş yarı-Einstein manifoldunun ilişkili skalerlerinin sağlaması gereken bazı şartlar elde edilmiştir. Daha sonra, genelleştirilmiş yarı-Einstein manifoldunun aynı zamanda pseudo Ricci simetrik olması durumunda, üreteç vektör alanları incelenmiş ve bu vektör alanları ile ilgili olarak çeşitli teoremler ispatlanmıştır. Çalışmanın beşinci ve son bölümünde ise, $(+,+,-,-)$, $(+,+,+,-)$ ve $(+,+,+,+)$ metrik işaretlerine sahip 4-boyutlu manifoldlar göz önüne alınmıştır. Bu manifoldlar üzerinde ikinci mertebeden simetrik ve reküran olan tensör alanları incelenmiştir. Genel yaklaşım, ikinci mertebeden simetrik tensörlerin söz konusu metrik işarete göre sınıflandırılmasına dayanmaktadır. Bu problemde ilk olarak, ikinci mertebeden simetrik ve paralel olan tensör alanları araştırılmıştır. Bu tensör alanları, ilk olarak, 4-boyutlu, nötr işaretli manifoldlar olarak isimlendirilecek olan, $(+,+,-,-)$ metrik işaretli manifoldlar için Segre tiplerine göre belirlenmiştir. Daha sonra, aynı metrik işarete göre, ikinci mertebeden simetrik reküran özelliğine sahip tensör alanları araştırılmıştır. Bu incelemeler, metriğin dolanım grubu göz önüne alınarak yapılmıştır. Daha sonra, ikinci mertebeden simetrik tensör alanları için, ilk problemde elde edilen sonuçlar özel olarak Ricci tensörüne uygulanmıştır. Ricci tensörünün paralel olması ve reküran olması problemi ele alınmıştır. Metrik işareti $(+,+,-,-)$ olan 4-boyutlu manifoldlar için Ricci tensörünün mümkün olabileceği bütün Segre tipleri ve dolanım tipleri elde edilmiştir. Ayrıca, bu uygulama sayesinde, bu manifoldun Einstein manifoldu olması durumu da incelenmiştir. Son olarak, $(+,+,-,-)$ metriği için ele alınan problemler, sırasıyla, Lorentz ve pozitif tanımlı metrik işaret olarak nitelendirilen, $(+,+,+,-)$ ve $(+,+,+,+)$ işaretli metriğe sahip manifoldlarda da araştırılmıştır. Benzer şekilde, dolanım teorisi göz önüne alınarak elde edilen sonuçlar Ricci tensörüne uygulanmıştır.
The purpose of this thesis is to examine the vector fields which can be defined on some geometrically and physically important manifolds and investigate their properties. By considering some vector fields on these special manifolds, various theorems related to these vector fields are proved and some geometrical properties of these manifolds are obtained. In the first chapter, it is mentioned about some manifolds and special vector fields which are taken into consideration in the study. For this purpose, first of all, a review of literature related to these special manifolds is given. Furthermore, the application areas of these special manifolds and vector fields to many other branches of science and engineering disciplines, mainly differential geometry and physics is highlighted. In the second chapter, it is introduced some fundamental concepts which are used in the study. Then, the definitions of some special vector fields such as torse-forming vector fields, concircular vector fields, recurrent vector fields, parallel vector fields, $\varphi(\rm Ric)$-vector fields, which will be examined on discussed manifolds, are given. In addition, some known results about these notions which will be considered later are given. In the third chapter, the types of the generalizations of Einstein manifolds are examined. In this examination, three kinds of these manifolds which are called the quasi-Einstein manifolds, the nearly quasi-Einstein manifolds and the generalized quasi-Einstein manifolds are considered. Firstly, the problem of examining some special vector fields on quasi-Einstein manifolds which are physically important is discussed. It is shown that if the generator vector field of the quasi-Einstein manifold is a $\varphi(\rm Ric)$-vector field, then this vector field must be parallel. Also, it is considered that if a quasi-Einstein manifold admits a $\varphi(\rm Ric)$-vector field, the relationship between the generator of the manifold and this vector field is found. On the other hand, an example of a quasi-Einstein manifold which is $4-$dimensional is given. Moreover, conformal mappings between quasi-Einstein manifolds admitting special vector fields are investigated. Under conformal mapping, some theorems are proved and some properties of $\varphi(\rm Ric)$-vector fields and concircular vector fields are obtained. At the same time, in the case the conformal transformation is also a conharmonic one, which is a conformal transformation preserving the harmonicity of a certain function, the relations between the associated scalars of quasi-Einstein manifolds are found. After that, in this chapter, nearly quasi-Einstein manifolds are considered. By considering a special type nearly quasi-Einstein manifold, the conditions of being special vector fields of the generators of this manifold are investigated and some theorems about this manifold are proved. In addition, conformal mappings of nearly quasi-Einstein manifolds admitting special vector fields are considered. Under the conformal transformation and conharmonic transformation, some results on concircular vector fields and $\varphi(\rm Ric)$-vector fields are given. Moreover, examples of these manifolds on $4-$dimensional spaces are given and nearly quasi-Einstein space-time is studied. In this chapter, finally, the problem of examining special vector fields on generalized quasi-Einstein manifolds is considered. The conditions for a generalized quasi-Einstein manifold admitting special vector fields when the Ricci tensor of the manifold satisfies some conditions are determined. Torse-forming vector fields and $\varphi(\rm Ric)$-vector fields are examined on these manifolds. Also, the conditions whether the generators, which are described as vector fields on these manifolds, can be some special vector fields or not are investigated. Meanwhile, under the certain specific conditions taken into the Ricci tensor, some properties of special vector fields defined on the manifold are examined. In the fourth chapter, pseudo Ricci symmetric and almost pseudo Ricci symmetric manifolds are considered. The properties of pseudo Ricci symmetric generalized quasi-Einstein manifolds and almost pseudo Ricci symmetric generalized quasi-Einstein manifolds are investigated. From the known results about pseudo Ricci symmetric manifolds, some conditions required to be provided by the associated scalars are obtained. Later, assuming that a generalized quasi-Einstein manifold is also a pseudo Ricci symmetric, the associated vector fields of this manifold are examined and some theorems related to these vector fields are proved. Some relations between the generator vector fields of the pseudo Ricci symmetric manifold and the generalized quasi-Einstein manifold are found. In this chapter, finally, by assuming that a generalized quasi-Einstein manifold is also an almost pseudo Ricci symmetric manifold, some relations between the generators are found likewise in the pseudo Ricci symmetric case. In the fifth and the last chapter, $4-$dimensional manifolds admitting a metric of signature $(+,+,-,-)$, $(+,+,+,-)$ and $(+,+,+,+)$ are considered. The recurrence structure of the second order symmetric tensors on these manifolds are examined. The technique used is to first solve the problem when the tensor in question is either parallel (covariantly constant) or can be scaled so that it is parallel. Then one considers those recurrent tensors which are not in this category. This will allow a definition of proper recurrence for such tensors (and the vector fields) and it is investigated in this work. The general approach is based on the classification scheme of the second order symmetric tensors with respect to the metric signature which is at issue. The analysis is based on the holonomy group of the Levi-Civita connection associated with the metric, the possible Lie algebras of which are known. In this problem, the tensor fields which are the second order symmetric and parallel are investigated initially. Firstly, these tensor fields are determined with respect to the Segre types for $4-$dimensional manifolds with signature $(+,+,-,-)$ which will be called a neutral signature. Then, the tensor fields which are the second order symmetric and recurrent are investigated for the same metric signature. These investigations are made by considering the holonomy group of the metric. After that, the results obtained for the second order symmetric tensors in the first problem are applied specially to the Ricci tensor. Thus, the problems of Ricci tensor which is parallel and Ricci-recurrent are examined. These problems are investigated by considering the possible Segre types for the Ricci tensor on the manifold. All possible Segre types and holonomy types for the Ricci tensor are obtained for $4-$dimensional manifolds admitting a metric with signature $(+,+,-,-)$. Also, with the help of this application, the case of being an Einstein manifold, that is the Ricci tensor is proportional to the metric tensor, of this manifold is studied. A by-product of this study is the finding of certain useful, direct techniques to study the properties of the various holonomy types using direct algebraic methods and the Ambrose-Singer theorem. Finally, the same problems discussed for the metric signature $(+,+,-,-)$ are investigated for the metrics of signature $(+,+,+,-)$ and $(+,+,+,+)$ which are called Lorentz signature and positive definite signature, respectively. Similarly, the results obtained by considering the holonomy theory are then applied to the Ricci tensor and the possible Segre types and holonomy types are determined for these metric signatures.
The purpose of this thesis is to examine the vector fields which can be defined on some geometrically and physically important manifolds and investigate their properties. By considering some vector fields on these special manifolds, various theorems related to these vector fields are proved and some geometrical properties of these manifolds are obtained. In the first chapter, it is mentioned about some manifolds and special vector fields which are taken into consideration in the study. For this purpose, first of all, a review of literature related to these special manifolds is given. Furthermore, the application areas of these special manifolds and vector fields to many other branches of science and engineering disciplines, mainly differential geometry and physics is highlighted. In the second chapter, it is introduced some fundamental concepts which are used in the study. Then, the definitions of some special vector fields such as torse-forming vector fields, concircular vector fields, recurrent vector fields, parallel vector fields, $\varphi(\rm Ric)$-vector fields, which will be examined on discussed manifolds, are given. In addition, some known results about these notions which will be considered later are given. In the third chapter, the types of the generalizations of Einstein manifolds are examined. In this examination, three kinds of these manifolds which are called the quasi-Einstein manifolds, the nearly quasi-Einstein manifolds and the generalized quasi-Einstein manifolds are considered. Firstly, the problem of examining some special vector fields on quasi-Einstein manifolds which are physically important is discussed. It is shown that if the generator vector field of the quasi-Einstein manifold is a $\varphi(\rm Ric)$-vector field, then this vector field must be parallel. Also, it is considered that if a quasi-Einstein manifold admits a $\varphi(\rm Ric)$-vector field, the relationship between the generator of the manifold and this vector field is found. On the other hand, an example of a quasi-Einstein manifold which is $4-$dimensional is given. Moreover, conformal mappings between quasi-Einstein manifolds admitting special vector fields are investigated. Under conformal mapping, some theorems are proved and some properties of $\varphi(\rm Ric)$-vector fields and concircular vector fields are obtained. At the same time, in the case the conformal transformation is also a conharmonic one, which is a conformal transformation preserving the harmonicity of a certain function, the relations between the associated scalars of quasi-Einstein manifolds are found. After that, in this chapter, nearly quasi-Einstein manifolds are considered. By considering a special type nearly quasi-Einstein manifold, the conditions of being special vector fields of the generators of this manifold are investigated and some theorems about this manifold are proved. In addition, conformal mappings of nearly quasi-Einstein manifolds admitting special vector fields are considered. Under the conformal transformation and conharmonic transformation, some results on concircular vector fields and $\varphi(\rm Ric)$-vector fields are given. Moreover, examples of these manifolds on $4-$dimensional spaces are given and nearly quasi-Einstein space-time is studied. In this chapter, finally, the problem of examining special vector fields on generalized quasi-Einstein manifolds is considered. The conditions for a generalized quasi-Einstein manifold admitting special vector fields when the Ricci tensor of the manifold satisfies some conditions are determined. Torse-forming vector fields and $\varphi(\rm Ric)$-vector fields are examined on these manifolds. Also, the conditions whether the generators, which are described as vector fields on these manifolds, can be some special vector fields or not are investigated. Meanwhile, under the certain specific conditions taken into the Ricci tensor, some properties of special vector fields defined on the manifold are examined. In the fourth chapter, pseudo Ricci symmetric and almost pseudo Ricci symmetric manifolds are considered. The properties of pseudo Ricci symmetric generalized quasi-Einstein manifolds and almost pseudo Ricci symmetric generalized quasi-Einstein manifolds are investigated. From the known results about pseudo Ricci symmetric manifolds, some conditions required to be provided by the associated scalars are obtained. Later, assuming that a generalized quasi-Einstein manifold is also a pseudo Ricci symmetric, the associated vector fields of this manifold are examined and some theorems related to these vector fields are proved. Some relations between the generator vector fields of the pseudo Ricci symmetric manifold and the generalized quasi-Einstein manifold are found. In this chapter, finally, by assuming that a generalized quasi-Einstein manifold is also an almost pseudo Ricci symmetric manifold, some relations between the generators are found likewise in the pseudo Ricci symmetric case. In the fifth and the last chapter, $4-$dimensional manifolds admitting a metric of signature $(+,+,-,-)$, $(+,+,+,-)$ and $(+,+,+,+)$ are considered. The recurrence structure of the second order symmetric tensors on these manifolds are examined. The technique used is to first solve the problem when the tensor in question is either parallel (covariantly constant) or can be scaled so that it is parallel. Then one considers those recurrent tensors which are not in this category. This will allow a definition of proper recurrence for such tensors (and the vector fields) and it is investigated in this work. The general approach is based on the classification scheme of the second order symmetric tensors with respect to the metric signature which is at issue. The analysis is based on the holonomy group of the Levi-Civita connection associated with the metric, the possible Lie algebras of which are known. In this problem, the tensor fields which are the second order symmetric and parallel are investigated initially. Firstly, these tensor fields are determined with respect to the Segre types for $4-$dimensional manifolds with signature $(+,+,-,-)$ which will be called a neutral signature. Then, the tensor fields which are the second order symmetric and recurrent are investigated for the same metric signature. These investigations are made by considering the holonomy group of the metric. After that, the results obtained for the second order symmetric tensors in the first problem are applied specially to the Ricci tensor. Thus, the problems of Ricci tensor which is parallel and Ricci-recurrent are examined. These problems are investigated by considering the possible Segre types for the Ricci tensor on the manifold. All possible Segre types and holonomy types for the Ricci tensor are obtained for $4-$dimensional manifolds admitting a metric with signature $(+,+,-,-)$. Also, with the help of this application, the case of being an Einstein manifold, that is the Ricci tensor is proportional to the metric tensor, of this manifold is studied. A by-product of this study is the finding of certain useful, direct techniques to study the properties of the various holonomy types using direct algebraic methods and the Ambrose-Singer theorem. Finally, the same problems discussed for the metric signature $(+,+,-,-)$ are investigated for the metrics of signature $(+,+,+,-)$ and $(+,+,+,+)$ which are called Lorentz signature and positive definite signature, respectively. Similarly, the results obtained by considering the holonomy theory are then applied to the Ricci tensor and the possible Segre types and holonomy types are determined for these metric signatures.
Açıklama
Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2016
Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2016
Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2016
Anahtar kelimeler
Manifold,
Ricci tensörü,
vektör alanı,
dolanım teorisi,
Manifold,
Ricci tensor,
vector field,
holonomy theory