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Öğe 2-Ruled hypersurfaces in Euclidean 4-space(Elsevier, 2021) Altin, Mustafa; Kazan, Ahmet; Yoon, Dae WonIn 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.Öğe 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, MustafaIn 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.Öğe 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, MustafaIn 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 PublishersÖğe CANAL HYPERSURFACES ACCORDING TO GENERALIZED BISHOP FRAMES IN 4-SPACE(Univ Nis, 2022) Kazan, Ahmet; Altin, MustafaIn 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.Öğe CANAL HYPERSURFACES GENERATED BY NON-NULL CURVES IN LORENTZ-MINKOWSKI 4-SPACE(Korean Mathematical Soc, 2023) Altin, Mustafa; Kazan, Ahmet; Yoon, Dae WonIn 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.Öğe Generalized rotation surfaces in E4 with density(Elsevier, 2023) Kazan, Ahmet; Altin, Mustafa; Yoon, Dae WonIn 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.Öğe 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, MustafaIn 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.Öğe Geometric characterizations of canal hypersurfaces in Euclidean spaces(Univ Nis, Fac Sci Math, 2023) Kazan, Ahmet; Altin, Mustafa; Yoon, Dae WonIn 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.Öğe LAPLACE-BELTRAMI MINIMALITY OF TRANSLATION HYPERSURFACES IN E4(Honam Mathematical Soc, 2023) Kazan, Ahmet; Altin, MustafaIn 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.Öğe ON CERTAIN SPACE CURVES DUE TO PARALLEL TRANSPORT FRAME IN En(Univ Nis, 2025) Singh, Charan; Kazan, Ahmet; Altin, Mustafa; Kazan, Sema; Jamali, MohammedIn 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.Öğe Ruled and Rotational Surfaces Generated by Non-Null Curves with Zero Weighted Curvature in (L3, ax2(Int Electronic Journal Geometry, 2020) Altin, Mustafa; Kazan, Ahmet; Karada, H. BayramIn this study, firstly we give the weighted curvatures of non-null planar curves in Lorentz-Minkowski space with density eax(2)+by(2) and obtain the planar curves whose weighted curvatures vanish in this space under the condition that the constants a and b are not zero at the same time. After giving the Frenet vectors of the non-null planar curves with zero weighted curvature in Lorentz-Minkowski space with density eax(2), we create the Smarandache curves of them. With the aid of these curves and their Smarandache curves, we get the ruled surfaces whose base curves are non-null curves of which vanishing weighted curvature and ruling curves are Smarandache curves of them. Followingly, we give some characterizations for these ruled surfaces by obtaining the mean and Gaussian curvatures, distribution parameters and striction curves of them. Also, rotational surfaces which are generated by non-null planar curves with zero weighted curvatures in Lorentz-Minkowski space E-1(3) with density eax(2) +by(2) are studied under the condition that the constants a and b are not zero at the same time. We draw the graphics of the obtained surfaces.Öğe Some classifications for Gauss map of Tubular hypersurfaces in E41 concerning linearized operators Gk(Hacettepe Univ, Fac Sci, 2025) Kazan, Ahmet; Altin, Mustafa; Turgay, Nurettin CenkIn this study, we deal with the Lauss map of tubular hypersurfaces in 4-dimensional Lorentz-Minkowski space concerninL the linearized operators L-1 (ChenL-Yau) and L-2. We obtain the L-1 (ChenL-Yau) operator of the Lauss map of tubular hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres or pseudo hyperbolic hyper-spheres whose centers lie on timelike or spacelike curves with non-null Frenet vectors in E( 1 )(4)and Live some classifications for these hypersurfaces which have Leneralized L(K )1-type Lauss map, first kindL(K-)pointwise 1-type Lauss map, second kindLK-pointwise 1-type Lauss map andL(K)-harmonic Lauss map, k E {1, 2}.Öğe TIMELIKE GENERAL ROTATIONAL SURFACES IN MINKOWSKI 4-SPACE WITH DENSITY(Korean Mathematical Soc, 2024) Altin, Mustafa; Kazan, Ahmet; Moon, Dae wonIn this study, we give weighted mean and weighted Gaussian curvatures of two types of timelike general rotational surfaces with non- null plane meridian curves in four-dimensional Minkowski space E 4 1 with density e lambda 1 x 2 +lambda 2 y 2 +lambda 3 z 2 +lambda 4 t 2 , where lambda i (i = 1, 2, 3, 4) are not all zero. We give some results about weighted minimal and weighted flat timelike general rotational surfaces in E 4 1 with density. Also, we construct some examples for these surfaces.
