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Let M be an n−dimensional differentiable manifold with a symmetric connection ∇ and T * M be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension g ∇,c on T * M defined by means of a symmetric (0, 2)-tensor field c on M. We get the conditions under which T * M endowed with the horizontal lift H J of an almost complex structure J and with the metric g ∇,c is a Kähler-Norden manifold. Also curvature properties of the Levi-Civita connection and another metric connection of the metric g ∇,c are presented.2000 Mathematics Subject Classification. Primary 53C07, 53C55; Secondary 53C22.

Abstract:In this study, we obtain surfaces at a constant distance from the edge of regression on a tubular surface indicated by M f , condition that M is denoted by a tubular surface in E 3 . Firstly, we show that M f is a tubular surface, for λ 1 = 0. Then, we calculate curvatures of M f and find some relationships between curvatures of surfaces M and M f . Finally, we research curvatures of center curve of M f , for some special cases.

ÖzetBu çalışmanın amacı televizyon, bilgisayar, kitap ve oyuncağın okul öncesi eğitime devam eden çocukların hayatındaki yeri ve önemini ortaya koymaktır. Çalışmada çocukların çizdikleri resimler ve görüşmelerden elde edilen veriler kullanılmıştır. Çalışmaya üç farklı özel okuldan okul öncesi eğitimine devam eden 5-6 yaş aralığında 51 çocuk katılmıştır. Yapılan çalışma iki aşamadan oluşmaktadır. Araştırmanın birinci aşamasında çocukların kendilerini televizyon, bilgisayar, kitap ve oyuncaklar ile resmetmesi istenmiştir. İkinci aşamada ise bu resimler üzerinden çocuklarla görüşmeler yapılmıştır. Ardından resimler ve bu resimler üzerinden yapılan görüşmeler analiz edilmiştir. Yapılan içerik analizi sonucu kategoriler belirlenmiştir. Araştırma sonuçlarına göre çocuklar en çok çizgi film seyretmekte ve popüler çizgi film karakterlerinin figürleriyle oynamayı tercih etmektedirler. Çocuklar bu uyaranlarla beraberken genel olarak olumlu duygular yaşamaktadırlar ancak kitap ve oyuncakla oynarken neler hissettiklerini tarif ederken daha güçlü ifadeler kullanmışlardır.Anahtar Kelimeler: Okul öncesi dönem, resim, televizyon, bilgisayar, kitap, oyuncak AbstractThe purpose of this study is to understand the place of four stimuli in lives of children attending early childhood; television, computer, books and toys. In the present study, data obtained from children's drawing and interviews was analyzed. Fifty-one children between the age of 5 and 6 participated in the study. They were attending three private kindergartens. First, the children were asked to draw themselves with a television, computer, books and toys.

Öz Bu çalışmada, ilk olarak Hasimoto yüzeyler ve paralel yüzeyler tanıtılmıştır. İkinci olarak, Hasimoto yüzeyler ve paralel yüzeylerle ilgili temel tanım ve teoremler verilmiştir. Bundan sonra, elde edilen paralel yüzeylerin birinci ve ikinci temel form katsayıları hesaplanmıştır. Böylece, Gauss ve ortalama eğrilikler bulunmuştur ve asıl yüzey ve paralelinin eğrilikleri arasındaki ilişkiler verilmiştir. Ayrıca, bu eğrilikleri kullanarak, bazı diferansiyel geometrik sonuçlar verilmiştir ve Hasimoto yüzeyi ve paralel yüzeyinin parametre eğrilerinin hangi şart altında geodezik, asimptotik veya eğrilik çizgisi olma durumları tartışılmıştır. Son olarak, eğer Hasimoto yüzeyini üreten eğri bir düzlem eğrisi ise o zaman paralel eğrinin eğriliği hesaplanmıştır. Bir örnek verilmiştir ve elde edilen eğriler Mathematica yardımıyla çizilmiştir.

In this paper, new types of associated curves, which are defined as rectifying-direction, osculating-direction, and normal-direction, in a three-dimensional Lie group G are achieved by using the general definition of the associated curve, and some characterizations for these curves are obtained. Additionally, connections between the new types of associated curves and the curves, such as helices, general helices, Bertrand, and Mannheim, are given. MSC: 53A04; 22E15 IntroductionMany authors have made significant contributions to the theory of curves from past to present. Some of these studies indicated that the relationships between the curvatures of the space curves are quite remarkable, and the new special curves are also defined via these relations [1][2][3][4][5]. Helices, one of these special space curves, have been studied by many researchers [6][7][8][9]. In addition to special space curves, some of the relationships between the curve pairs are also particularly interesting. The curve pairs are obtained by using the Frenet vectors or curvatures. In this respect, involute-evolute, Bertrand, and Mannheim curves are well-known examples of curve pairs, and many studies have been performed on this topic [10][11][12][13][14][15][16].The Riemannian geometry of a Lie group was studied in [17]. Here, the rich collection of examples that are obtained by providing an arbitrary Lie group G with a Riemannian metric invariant under left translations was given. The semi-Riemannian geometry of a Lie group was examined in [18]. They also obtained the sectional curvature in terms of Lie invariants based on the semisimple case. Furthermore, the curves mentioned above have been handled in Lie group theory by many authors [14,[19][20][21][22].In [23], the authors explained the notions of both the principal (binormal)-direction curve and principal (binormal)-donor curve of a Frenet curve in E 3 . They characterized some special curves in E 3 by using the relationships between the curves.In this study, within the framework of the definition of associated curves, we introduce new types of direction curves in a three-dimensional Lie group G, and we characterize these curves. Finally, we determine the relationships between the new types of direction curves (rectifying-direction, osculating-direction, and normal-direction curve curves) and the curves (Bertrand curve, involute-evolute, rectifying curve, etc.).

In this paper, the adjoint curve is defined by using the alternative moving frame of a unit speed space curve in 3-dimensional Euclidean space. The relationships between Frenet vectors and alternative moving frame vectors of the curve are used to offer various characterizations. Besides, ruled surfaces are constructed with the curve and its adjoint curve, and their properties are examined. In the last section, there are examples of the curves and surfaces defined in the previous sections.

In this paper, we investigate surface at a constant distance from the edge of regression on a surface of revolution indicated by , condition that is denoted by a surface of revolution in 3. First of all, we show that is a surface of revolution. Furthermore, we calculate curvatures of and obtain some relationships between curvatures of surfaces and for some special cases.

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