Özel singüler eğrilerin geometrisi üzerine
On the geometry of special singular curves
Yazar: BAHAR DOĞAN YAZICI
Danışman: PROF. DR. MURAT TOSUN
Sakarya Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Ana Bilim Dalı / Geometri Bilim Dalı
Derece: Doktora, 2022
Konu: Matematik = Mathematics
Dizin: Bertrand eğrileri = Bertrand curves ; Eğriler = Curves ; Regle yüzeyler = Regle surfaces ; Öklid uzayları = Euclidean spaces

Özet:
Bu tez altı bölümden oluşmaktadır. Birinci bölüm giriş bölümü olup, literatür incelemeleri verilmiştir.
İkinci bölümde Öklidyen uzayda eğri teori, yüzey teori ve temel singülerite teori için temel tanım ve teoremlere yer verilmiştir. Üçüncü bölümde özel bir singüler eğri olan framed eğriler tanıtılmıştır ve ayrıntılı olarak temel tanım ve teoremler verilmiştir. Framed eğrilerin varlık ve teklik koşulları, üç boyutlu Öklidyen uzayda framed eğriler, framed eğrilerin Bishop-tipi çatısı, framed eğrilerin Frenet-tipi çatısı ve framed eğrilerin Sabban-tipi çatısı verilmiştir. Ayrıca singüler bir yüzey olabilen framed yüzey teorisi tanıtılmıştır. Framed yüzeylerin temel değişmezleri, framed yüzeylerin eğrilikleri verilmiştir. Bu yapılar yardımıyla framed yüzeylerin cuspidal edge, swallowtail ve cuspidal cross cap yapılarına lokal difeomorfik olma şartları incelenmiştir. Dördüncü bölümden itibaren tezin orjinal kısmı başlamaktadır. Dördüncü bölümde özel framed düzlem eğrileri tanıtılmıştır. İlk olarak framed normal eğriler ve karakterizasyonları verilmiştir. Bir framed normal eğrinin framed küresel eğri olması için gerek ve yeter koşullar ifade edilmiştir. Daha sonra framed küresel eğrilerin genişlemesi yardımıyla framed rektifiyan eğriler incelenmiştir. Her framed küresel eğri ile oluşturulan genişlemenin bir framed rektifiyan eğri olmayabileceği gösterilmiştir. Framed küresel eğri ile oluşturulan genişleme sonucu elde edilen framed rektifiyan eğrinin framed vektörleri ve framed eğrilikleri verilmiştir. Beşinci bölümde framed eğriler için bir diferansiyel denklem tanıtılmıştır. Bu diferansiyel denklemin yapısı hakkında bazı önermeler ve sonuçlar verilmiştir. Ayrıca, elde edilen diferansiyel denklemin dördüncü bölümde tanıtılan framed normal eğri, framed rektifiyan eğriler ile arasındaki ilişkiler incelenmiştir. Diğer yandan, bu diferansiyel denklem sayesinde regüler helis eğrilerinin bir genellemesi olan framed helisler için yeni bir karakterizasyon verilmiştir. Altıncı bölümde, özel bir integral eğrisi olan framed eğrilerin adjoint eğrileri tanımlanmıştır. İlk olarak framed eğrilerin genel çatısına göre adjoint eğriler verilmiştir. Daha sonra framed eğrilerin Frenet-tipi çatısına göre adjoint eğriler verilmiştir. Framed temel eğri ve adjoint eğrisi arasındaki ilişkiler açıklanmıştır. Bir framed adjoint eğrisinin framed Bertrand ve framed Mannheim eğrisi olması için gerek ve yeter koşullar ifade edilmiştir. Ayrıca, dayanak eğrisi bir framed eğri, doğrultaman vektörü ise regüler bir adjoint eğri olan regle yüzeyler tanıtılmıştır. Son olarak, framed adjoint eğrilerinin özel yüzeyleri olan normal yüzeyler, binormal yüzeyler ve genelleştirilmiş normal yüzeyler verilmiştir. Framed adjoint eğrilerinin normal, binormal ve genelleştirilmiş normal yüzeylerinin silindirik olma, açılabilir olma koşulları incelenmiştir ve striksiyon eğrileri verilmiştir. Ayrıca bu yüzeylerin tüm geometrik yapılarının yanı sıra singülerite tipleri incelenmiştir. İlk olarak framed eğrilerin adjoint eğrilerinin normal, binormal ve genelleştirilmiş normal yüzeylerinin singüler nokta kümeleri oluşturulmuştur ve framed yüzey teorisi yardımıyla singüler noktalarının dejenere olmayan bir singüler nokta olması için koşullar verilmiştir. Daha sonra framed eğrilerin adjoint eğrilerinin normal, binormal ve genelleştirilmiş normal yüzeylerinin cross-cap, cuspidal edge, swallowtail ve cuspidal cross cap yapılarına lokal difeomorfik olma koşulları elde edilmiştir.

Abstract:
This thesis consists of six chapters. The first chapter is the introduction and literature reviews are given. In the second chapter, basic definitions and theorems for curve theory, surface theory and basic singularity theory in Euclidean space are given. For the theory of curves in Euclidean geometry, basic definitions such as regular curve, unit speed curve, Frenet- Serret formulas, integral curves, helices, Bertrand curves, Mannheim curves, Bishop frame, spherical curves, Sabban frame are given. For the theory of surfaces in Euclidean geometry, basic definitions such as patch, regular surface, surface normal, singular points of the surface, shape operator, Gaussian curvature, mean curvature, ruled surfaces, distribution parameter, developable ruled surfaces, cylindrical surfaces, cone surfaces, striction points, striction curve, normal surfaces, binormal surfaces, generalized normal surfaces are given. For the basic singularity theory in Euclidean geometry, definitions of ordinary cusp singularity in the plane, types of singular points in the plane, ordinary inflection singularity, Ak−singularity, cuspidal-edge, swallowtail and cuspidal cross cap are given. In the third chapter, framed curves, which are a special singular curve, are introduced and basic definitions and theorems are given in detail. First, the definition of a framed curve in an n -dimensional Euclidean space is given. Existence and uniqueness theorems of framed curves are expressed by using the existence and uniqueness conditions of differential equations. Then, framed curves in three-dimensional Euclidean space, Bishop-type frame of framed curves, Frenet-type frame of framed curves and Sabban-type frame of framed curves are given. Also, the theory of framed
surface, which can be a singular surface, is introduced. Fundamental invariants of framed surfaces and curvatures of framed surfaces are given. With the help of these structures, the local diffeomorphic conditions of the cuspidal edge, swallowtail and cuspidal cross cap structures of the framed surfaces were investigated. From the fourth chapter, the original part of the thesis begins. In the fourth chapter, special framed plane curves are introduced. First, framed normal curves and their characterizations are given. Framed normal curves are considered as curves lying in the plane formed by the generalized normal and generalized binormal vectors of the position vectors of the framed curves. Some equations describing the relationships between framed curvatures of framed normal curves have been obtained. The behavior of these equations at regular and singular points has been investigated. The fact that a regular normal curve is a regular spherical curve, and a regular spherical curve is a regular normal curve has led to the investigation of the relationship between framed normal curves and framed spherical curves. Necessary and sufficient conditions are
expressed for a framed normal curve to be a framed spherical curve. An important finding is that given a framed curve with at least one non-singular point, the necessary and sufficient condition for the curve to be a framed normal curve is a framed spherical curve. Then, the framed rectifying curves were examined with the help of the dilation of the framed spherical curves. First, some theorems are stated for a framed curve to be a framed rectifying curve. The conditions that the framed curvature of the framed rectifying curves must meet are reminded. As a method, a moving frame was first created for framed spherical curves. Then, by using the frame elements obtained for framed spherical curves, the relationship with framed rectifying curves was examined. It has been shown that the dilation created by each framed spherical curve may not be a framed rectifying curve. Framed vectors and framed curves of the framed rectifying curve obtained because of the dilation created by the framed spherical curve are given. In the fifth chapter, a differential equation for framed curves is introduced. First, the squared distance functions of the framed curves are introduced and a new function h= is given by using the distance squared function. A general differential equation is given for framed curves by using the squared distance function and the function h obtained from the distance squared function. Some propositions and conclusions about the structure of this differential equation are given. The general differential equation given for framed curves is a fourth-order differential equation with respect to the differential equation distance function if the curve is a regular framed curve, if the curve is a singular framed curve, it is a third-order differential equation with respect to the differential equation the function h . In addition, the relationship between the obtained differential equation and the framed normal curve, which was introduced in the fourth chapter, and the framed rectifying curves were examined. On the other hand, thanks to this differential equation, a new characterization is given for framed helices, which is a generalization of regular helix curves. The characterization given for framed helices in the literature is only a characterization based on their generalized curvature. In this thesis, the
characterization created by using a general differential equation for framed curves is a characterization that includes the function that determines the singular points of the framed curve. At the end of the chapter, some examples supporting general differential equation theories for framed curves, theorems examining the relationship of framed
normal and framed rectifying curves with this differential equation, and newly created characterizations for framed helixes are given. In the sixth chapter, adjoint curves of framed curves, which are a special integral curve, are defined. First, adjoint curves are given according to the general frame of framed curves. The relationships between the adjoint curves created according to the general frame and the framed vectors and framed curvatures of the framed curves are examined. Necessary and sufficient conditions are expressed for a framed adjoint curve to be a framed Bertrand and framed Mannheim curve. Then, adjoint curves are given according to the Frenet-type frame of the framed curves. The relationships between the framed base curve and the adjoint curve are explained. The relationships between the adjoint curves created according to the Frenet-type frame and the framed vectors and framed curvatures of the Frenet-type framed curves are investigated. It has been concluded that the adjoint curves formed according to the Frenet-type frame cannot be a planar curve, and if the Frenet-type framed curve is a framed helix, the adjoint curve is also a framed helix. In addition, ruled surfaces, whose base curve is a framed curve and whose direction vector is a regular adjoint curve, are introduced. With this ruled surface, some geometric characterizations such as being developable and cylindrical are included. Moreover, normal surfaces, binormal surfaces and generalized normal surfaces, which are the special surfaces of framed adjoint curves, are given. Firstly, the normal surface M1 is introduced which the base curve is an adjoint curve and direction is the generalized normal vector of the adjoint curve. The conditions for the surface M1 to be developable have been obtained. It has been stated and proven that the surface M1 is not a cylindrical surface. Conditions are expressed so that the striction curve of the surface 1M is the adjoint curve, which is the base curve. In addition, all geometric structures of these surfaces as well as singularity types were examined. First, the singular point set of the surface is expressed as two separate sets, but using the general structure of framed curves, it is shown that it has only one singular set of points. Then, the cross-cap conditions of the surface M1 are given for
the created singular point set. Later, the surface is considered as a framed surface since it has singular points. Using the framed surface theory, a moving frame of the surface M1 is constructed. In addition, the curvatures of the surface are expressed. After showing that the singular points of the surface M1 are non-degenerate singular points, the cuspidal-edge, swallowtail and cuspidal cross cap structures of the surface were given local diffeomorphic conditions. Secondly, the binormal surface M2 is introduced which the base curve is an adjoint curve and direction is the generalized binormal vector of the adjoint curve. Similar to the surface M1 , the geometric structure of the surface M2 was examined and singularity types were determined. Also, the singular points of the surface M2 were shown to be non-degenerate singular points, but the surface proved not locally diffeomorphic to a cuspidal-edge, swallowtail, and
cuspidal cross cap structures. Finally, the base curve, an adjoint curve, and the directrix of the generalized normal and generalized binormal vectors, the generalized normal surface M3 is introduced. It is clearly stated that the generalized normal surface M3 is a generalization of both the normal surface M1 and the binormal surface M2. The singular point sets of the surface 3 M were formed as two separate sets. It is shown that, unlike the M1 and M2 surfaces, the surface 2 M has two sets of singular points. Moreover, necessary and sufficient conditions are expressed for the surface M3 to be cross cap with respect to two separate singular point sets. After showing that the singular points are non-degenerate singular points for two separate singular point sets of the surface M3, the conditions for local diffeomorphic to the cuspidal edge, swallowtail and cuspidal cross cap structures are explained. At the end of the chapter, a Frenet-type framed curve is given and a regular adjoint curve and a singular adjoint curve are formed. Then, normal surface, binormal surface and generalized normal surface, whose base curve is a regular adjoint curve, are introduced. In addition, normal surface, binormal surface and generalized normal surface, whose base curve is a singular adjoint curve, are given. Therefore, some figures are given for the six ruled surfaces created. Therefore, this example has been an example that supports the theory.