İzotropik submersiyonlar üzerine
Küçük Resim Yok
Tarih
2023
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Ege Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Bu tez altı bölümden oluşmaktadır. Birinci bölümde submersiyonlar, Riemann submersiyonlar kavramlarının ortaya çıkış süreci ve hangi amaçlar ile kullanıldığına dair kısa bilgiler verilmiştir ve bu tezdeki diğer bölümlerin içeriklerinden kısaca bahsedilmiştir. İkinci bölümde tez boyunca kullanılacak olan temel tanım ve teoremler verilmiştir. Üçüncü bölümde submersiyon ve Riemann submersiyon kavramları tanımlanmıştır. O'Neill'in temel tensörleri verilmiştir ve temel özellikleri incelenmiştir. Dördüncü bölümde temel tensörlerin kovaryant türevleri verilerek manifoldların eğrilikleri arasındaki bağıntılar elde edilmiştir. Beşinci bölümde, 2020 yılında tanımlanan izotropik Riemann submersiyon kavramına, temel manifold ve hedef manifoldun eğrilikleri arasındaki ilişkiye izotropik Riemann submersiyon kavramı altında bakılmıştır. Ayrıca noktasal düzlemsel yatay kesitlerle izotropik submersiyon kavramı arasındaki ilişki incelenmiştir. Altıncı bölümde, tezin orijinal kısmının başlangıcı olup sabit eğrilikli izotropik Riemann submersiyon kavramı tanıtılıp temel ve hedef manifoldun geometrisi incelenmiştir. Ayrıca izotropik Riemann submersiyonlar A temel tensörü kullanılarak tanımlanmıştır ve bu submersiyonlara yatay izotropik Riemann submersiyon adı verilmiştir. Yatay izotropik Riemann submersiyon yardımıyla temel ve hedef manifoldun geometrisi incelenip bu manifoldların eğrilikleri arasındaki ilişkiler araştırılmıştır.
This thesis consists of six chapters. In the first chapter, brief information is given about the emergence process of the concepts of submersions, Riemann submersions and for what purposes they are used, and the contents of the other chapters in this thesis are briefly mentioned. In the second chapter, the basic definitions and theorems that will be used throughout the thesis are given. In the third chapter, the concepts of submersion and Riemann submersion are defined. O'Neill's fundamental tensors are given and their basic properties are examined. In the fourth chapter, the covariant derivatives of the fundamental tensors are given and the relations between the curvatures of the manifolds are obtained. In the fifth chapter, the concept of isotropic Riemann submersion, defined in 2020, and the relationship between the curvatures of the base manifold and the target manifold are examined under the concept of isotropic Riemann submersion. In addition, the relationship between point planar horizontal sections and the concept of isotropic submersion is examined. In the sixth chapter, which is the beginning of the original part of the thesis, the concept of isotropic Riemann submersion with constant curvature is introduced and the geometry of the base and target manifold is examined. In addition, isotropic Riemann submersions are defined using the principal tensor A and these submersions are called horizontal isotropic Riemann submersions. With the help of horizontal isotropic Riemann submersion, the geometry of the base and target manifolds were examined and the relationships between the curvatures of these manifolds were investigated.
This thesis consists of six chapters. In the first chapter, brief information is given about the emergence process of the concepts of submersions, Riemann submersions and for what purposes they are used, and the contents of the other chapters in this thesis are briefly mentioned. In the second chapter, the basic definitions and theorems that will be used throughout the thesis are given. In the third chapter, the concepts of submersion and Riemann submersion are defined. O'Neill's fundamental tensors are given and their basic properties are examined. In the fourth chapter, the covariant derivatives of the fundamental tensors are given and the relations between the curvatures of the manifolds are obtained. In the fifth chapter, the concept of isotropic Riemann submersion, defined in 2020, and the relationship between the curvatures of the base manifold and the target manifold are examined under the concept of isotropic Riemann submersion. In addition, the relationship between point planar horizontal sections and the concept of isotropic submersion is examined. In the sixth chapter, which is the beginning of the original part of the thesis, the concept of isotropic Riemann submersion with constant curvature is introduced and the geometry of the base and target manifold is examined. In addition, isotropic Riemann submersions are defined using the principal tensor A and these submersions are called horizontal isotropic Riemann submersions. With the help of horizontal isotropic Riemann submersion, the geometry of the base and target manifolds were examined and the relationships between the curvatures of these manifolds were investigated.
Açıklama
Anahtar Kelimeler
Matematik, Mathematics
