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2-Ruled hypersurfaces in Euclidean 4-space
(Elsevier, 2021) Altin, Mustafa; Kazan, Ahmet; Yoon, Dae Won
In this paper, firstly we obtain the Gauss map (unit normal vector field) of a 2-ruled hypersurface in Euclidean 4-space with the aid of its general parametric equation. After, we obtain Gaussian and mean curvatures of the 2-ruled hypersurface and give some characterizations about its minimality. Finally, we deal with the first and second Laplace-Beltrami operators of 2-ruled hypersurfaces in E-4 and give some results about them. (C) 2021 Elsevier B.V. All rights reserved.
A GEOMETRIC APPLICATION OF SOLITON SURFACES ASSOCIATED WITH THE BETCHOV-DA RIOS EQUATION USING AN EXTENDED DARBOUX FRAME FIELD IN E4
(Pergamon-Elsevier Science Ltd, 2025) Kazan, Ahmet; Altin, Mustafa
In this paper, for a soliton surface Omega = Omega (u, v) associated with the Betchov-Da Rios equation, we obtain the derivative formulae of an extended Darboux frame field of a unit speed u-parameter curve Omega = Omega(u, v) for all v. Also, we get the geometric invariants k and h of the soliton surface Omega = Omega(u, v) and we obtain the Gaussian curvature, mean curvature vector and Gaussian torsion of Omega. We give some important geometric characterizations such as flatness, minimality and semi-umbilicaly with the aid of these invariants. Additionally, we study the curvature ellipse of the Betchov-Da Rios soliton surface and Wintgen ideal (superconformal) Betchov-Da Rios soliton surface with respect to an extended Darboux frame field. Finally, we construct an application for the Betchov-Da Rios soliton surface with the aid of an extended Darboux frame field.
A Geometric Application of Soliton Surfaces Associated with The Betchov–Da Rios Equation Using an Extended Darboux Frame Field in E4
(Elsevier Ltd, 2025) Kazan, Ahmet; Altin, Mustafa
In this paper, for a soliton surface Ω = Ω(u, v) associated with the Betchov–Da Rios equation, we obtain the derivative formulae of an extended Darboux frame field of a unit speed u-parameter curve Ω = Ω(u, v) for all v. Also, we get the geometric invariants k and h of the soliton surface Ω = Ω(u, v) and we obtain the Gaussian curvature, mean curvature vector and Gaussian torsion of Ω. We give some important geometric characterizations such as flatness, minimality and semi-umbilicaly with the aid of these invariants. Additionally, we study the curvature ellipse of the Betchov–Da Rios soliton surface and Wintgen ideal (superconformal) Betchov–Da Rios soliton surface with respect to an extended Darboux frame field. Finally, we construct an application for the Betchov–Da Rios soliton surface with the aid of an extended Darboux frame field. © 2025 Polish Scientific Publishers
CANAL HYPERSURFACES ACCORDING TO GENERALIZED BISHOP FRAMES IN 4-SPACE
(Univ Nis, 2022) Kazan, Ahmet; Altin, Mustafa
In the present paper, we study canal hypersurfaces according to generalized Bishop frames of type B (parallel transport frame), type C and type D in Euclidean 4-space and obtain Gaussian, mean and principal curvatures of them in general form. We give some results for their flatness, minimality and we examine the Weingarten canal hypersurfaces according to these frames. Especially, we investigate the tubular hypersurfaces by taking the radius function is constant in these canal hypersurfaces.
CANAL HYPERSURFACES ACCORDING TO ONE OF THE EXTENDED DARBOUX FRAME FIELD IN EUCLIDEAN 4-SPACE
(Vinca Inst Nuclear Sci, 2022) Kazan, Ahmet
In the present study, we deal with canal hypersurfaces according to extended Dar-boux frame field of second kind in Euclidean 4-space (E-4) and in this context, firstly we obtain the Gaussian, mean and principal curvatures of the canal hypersurface according to extended Darboux frame field of second kind and give some results for flatness and minimality of these hypersurfaces in E-4. Also, we give some results for Weingarten canal hypersurfaces according to extended Darboux frame field of second kind in E-4 and finally, we construct an example.
CANAL HYPERSURFACES GENERATED BY NON-NULL CURVES IN LORENTZ-MINKOWSKI 4-SPACE
(Korean Mathematical Soc, 2023) Altin, Mustafa; Kazan, Ahmet; Yoon, Dae Won
In the present paper, firstly we obtain the general expression of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres, pseudo hyperbolic hyperspheres and null hypercones whose centers lie on a non-null curve with non-null Frenet vector fields in E14 and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures, mean curvatures and principal curvatures. Also, we give some results about their flatness and minimality conditions and Weingarten canal hypersurfaces. Also, we obtain these characterizations for tubular hypersurfaces in E-1(4) by taking constant radius function and finally, we construct some examples and visualize them with the aid of Mathematica.
Conformally-projectively flat trans-Sasakian statistical manifolds
(Elsevier, 2019) Kazan, Ahmet
In this study, the notion of trans-Sasakian statistical manifold is defined, some results for this manifold's curvature tensors are given, an example of 3-dimensional trans-Sasakian statistical manifold is constructed and some characterizations about ?-conformal-projective flatness of a trans-Sasakian statistical manifold are given.
e^(ax+by) Yoğunluklu E^3_1 Uzayında Sıfır Ağırlıklı Eğriliğe Sahip Null Olmayan Düzlemsel Eğrilerin Oluşturduğu Yüzeyler
(Erzincan Üniversitesi, 2020) Altın, Mustafa; Kazan, Ahmet
Bu çalışmada, ... yoğunluklu ... 1 3 Lorentz-Minkowski uzayında, ikisi aynı anda sıfır olmayan ... ve ... sabitlerinin durumlarına göre, ağırlıklı eğrilikleri sıfır olan spacelike ve timelike düzlemsel eğriler yardımıyla oluşturulan dönel yüzeyler ve regle yüzeyler çalışılmıştır.
Embankment Surfaces in Euclidean 3-Space and Their Visualizations
(Communications in Mathematics and Applications, 2019) Kazan, Ahmet; Karadağ, Hacı Bayram
In the present paper, we obtain the parametric representation of an embankment surface and give an example for it. We define the notions of embankmentlike surfaces and tubembankmentlike surfaces. Furthermore, we create some embankmentlike and tubembankmentlike surface examples with the aid of different directrix and draw these directrix and surfaces. Also, we find the Gaussian, mean and second Gaussian curvatures of these surfaces and draw the Gaussian, mean and second Gaussian curvature functions' graphics and the variations of Gaussian, mean and second Gaussian curvatures on related surfaces with the aid of Mathematica.
Generalized rotation surfaces in E4 with density
(Elsevier, 2023) Kazan, Ahmet; Altin, Mustafa; Yoon, Dae Won
In the present paper, we obtain the weighted mean and weighted Gaussian curvatures of a generalized rotation surface in E4 with density e lambda 1x2+lambda 2y2+lambda 3z2+lambda 4t2, where lambda i, i is an element of {1, 2, 3, 4}, are not all zero and we give some results about weighted minimal and weighted flat generalized rotation surface in E4 with density e lambda(x2+y2)+mu(z2+t2), where lambda and mu are not all zero. Also, we obtain the weighted mean and weighted Gaussian curvatures of Klein bottle, Banchoff surface, Vranceanu rotation surface, and Clifford torus, which are examples of generalized rotation surfaces, in E4 with density e lambda(x2+y2)+mu(z2+t2) and give some results for their minimality and flatness.(c) 2023 Elsevier B.V. All rights reserved.
Geometric Analysis of Soliton Surfaces via Generalized Bishop Frame Field of Type-C Associated with Betchov-Da Rios Equation in E4
(Mdpi, 2026) Li, Yanlin; Kazan, Ahmet; Altin, Mustafa
In this study, using the generalized Bishop frame field of type-C in four-dimensional Euclidean space, we investigate the geometric properties of a soliton surface M=M(x,y) associated with the Betchov-Da Rios equation. Using a unit speed x-parameter curve M=M(x,y), for all y, we derive the derivative formulas for the generalized Bishop frame field of type-C. We obtain two fundamental geometric invariants of the soliton surface, k and h, characterizing the points of the surface, as well as a few other significant invariants, including Gaussian curvature, a mean curvature vector and Gaussian torsion. We use these surface invariants to prove a collection of theorems that describe the circumstances in which the soliton surface is flat, minimal, semi-umbilic or Wintgen ideal (superconformal). Additionally, we provide a theorem that describes the B-DR soliton surface's curvature ellipse in relation to the generalized Bishop frame field of type-C in E4. Finally, we construct a foundational example to support our theoretical results and demonstrate the construction of the generalized Bishop frame field of type-C.
Geometric characterizations of canal hypersurfaces in Euclidean spaces
(Univ Nis, Fac Sci Math, 2023) Kazan, Ahmet; Altin, Mustafa; Yoon, Dae Won
In the present paper, firstly we obtain the general expression of canal hypersurfaces in Euclidean n-space and deal with canal hypersurfaces in Euclidean 4-space E4. We compute Gauss map, Gaussian curvature and mean curvature of canal hypersurfaces in E4 and obtain an important relation between the mean and Gaussian curvatures as 3H rho = K rho 3 - 2. We prove that, the flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones and minimal canal hypersurfaces are only generalized catenoids. Also, we state the expression of tubular hypersurfaces in Euclidean spaces and give some results about Weingarten tubular hypersurfaces in E4.
Helix Surfaces in Euclidean 3-Space with Density
(Osman SAĞDIÇ, 2021) Kazan, Ahmet; Kazan, Sema
The differential geometry of helix curves and helix hypersurfaces in different spaces has important application areas in many disciplines. Also, the notion of weighted manifold is become to be a very popular topic for scientists in recent years. In this context, after defining the notions of weighted mean curvature (or...-mean curvature) and weighted Gaussian curvature (or ...-Gaussian curvature) of an n-dimensional hypersurface on manifolds with density, lots of studies have been done by differential geometers in different spaces with different densities. So, in the present study, firstly we give the normal vector field, mean curvature and Gaussian curvature of a helix surface in three dimensional Euclidean space and after that, we obtain the weighted mean curvature and weighted Gaussian curvature of a helix surface generated by a unit speed planar curve in three dimensional Euclidean space with different three densities by stating the parametric equation of this surface. However, we know that a hypersurface is weighted minimal and weighted flat in Eucilidean 3-space with density if the weighted mean curvature and the weighted Gaussian curvature vanish, respectively. So, by using these definitions, we obtain the weighted minimal helix surfaces for these different densities and give some results for weighted flatness of the helix surfaces in Euclidean 3-space. We hope that, this study will bring a new viewpoint to differential geometers who are dealing with constant angle surfaces and in near future, one can handle these surfaces in different spaces with another densities.
Hypersurface families with Smarandache curves in Galilean 4-space
(2021) ALTIN, Mustafa; Kazan, Ahmet; Karadağ, H. Bayram
In this paper, we study the hypersurface families with Smarandache curves in 4-dimensional Galilean space G4 and give the conditions for different Smarandache curves to be parameter and the curve which generates the Smarandache curves is geodesic on a hypersurface in G4. Also, we investigate three types of marching-scale functions for one of these hypersurfaces and construct an example for it.
k-TYPE SLANT HELICES DUE TO GENERALIZED BISHOP FRAME (GBF) OF TYPE C AND TYPE D IN E4
(Palestine Polytechnic University, 2025) Singh, Charan; Kazan, Ahmet; Altın, Mustafa; Kazan, Sema; Jamali, Mohammed
The present article deals with the study of k-type slant helices with two equivalence classes of generalized Bishop frames (GBF) called type C frame and type D frame in E4 and obtain characterizations for such curves with these frames. © Palestine Polytechnic University-PPU 2025.
LAPLACE-BELTRAMI MINIMALITY OF TRANSLATION HYPERSURFACES IN E4
(Honam Mathematical Soc, 2023) Kazan, Ahmet; Altin, Mustafa
In the present paper, we study translation hypersurfaces in E4. In this context, firstly we obtain first, second and third Laplace-Beltrami (LBI, LBII and LBIII) operators of the translation hypersurfaces in E4. By solving second and third order nonlinear ordinary differential equations, we prove theorems that contain LBI-minimal, LBII-minimal and LBIII-minimal translation hypersurfaces in E4.
Lorentz-Minkowski 4-Uzayında Timelike Eğriler Tarafından Oluşturulan Tubular Hiperyüzeyler İçin Bazı Sınıflandırmalar
(2025) Kazan, Ahmet; ALTIN, Mustafa
Bu çalışmada, 4-boyutlu Lorentz-Minkowski uzayı E_1^4’de timelike eğriler tarafından oluşturulan tubular hiperyüzeylerin Gauss dönüşümlerinin doğrusallaştırılmış operatörü L_0 ile ilgilendik. Bu bağlamda ilk olarak, E_1^4’de timelike eğriler tarafından oluşturulan tubular hiperyüzeylerin Gauss dönüşümlerinin L_0 operatörünü elde ettik. Ardından sırasıyla, genelleştirilmiş L_0 1-tipli Gauss dönüşümüne sahip, birinci ve ikinci çeşit L_0-noktasal 1-tipli Gauss dönüşümüne sahip ve L_0-harmonik Gauss dönüşümüne sahip olan bu hiperyüzeyler için bazı sınıflandırmalar verdik.
Loxodromes on twisted surfaces in Lorentz-Minkowski 3-space
(World Scientific Publ Co Pte Ltd, 2023) Kazan, Ahmet; Altm, Mustafa
In this paper, first, we give the general formulas according to first fundamental form of a surface for different types of loxodromes, meridians and surfaces in E-1(3). After that, we obtain the differential equations of loxodromes on Type-I, Type-II and Type-III twisted surfaces in E-1(3) and also, we state a theorem which generalizes the differential equations of different types of loxodromes on the twisted surfaces for a special case. Finally, we provide several examples for visualizing our obtained results and draw our loxodromes and meridians on twisted surfaces with the aid of Mathematica.
Monge hypersurfaces in euclidean 4-space with density
(Gazi Üniversitesi, 2020) Altın, Mustafa; Kazan, Ahmet; Karadağ, Hacı Bayram
In the present study, firstly we give the mean and Gaussian curvatures of a Monge hypersurface in 4-dimensional Euclidean space. After this, we study on Monge hypersurfaces in with different densities. In this context, we obtain the weighted minimal and weighted flat Monge hypersurfaces in with densities (linear density) and with the aid of different choices of constants and , where and are not all zero constants.
ON CERTAIN SPACE CURVES DUE TO PARALLEL TRANSPORT FRAME IN En
(Univ Nis, 2025) Singh, Charan; Kazan, Ahmet; Altin, Mustafa; Kazan, Sema; Jamali, Mohammed
In this paper, we study k-type (k is an element of {0, 1, 2, ...., n-1}) slant helices due to non-zero parallel transport frame in E-n. We give some characterizations for 0-type, 1-type,..., and in general (n-1)-type slant helix due to parallel transport frame in terms of parallel transport curvatures in E-n and with the aid of these characterizations we give an important general theorem which gives the necessary and sufficient condition for any space curve to be k-type slant helices due to parallel transport frame in E-n. We also obtain the characterizations of the curves whose position vectors belong to the normal, rectifying and osculating spaces (called normal, rectifying and osculating curves, respectively) due to parallel transport frame in E-n.

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