In this paper, we first study invariants of curves parametrized by a real variable in the dual plane 𝔻^2 under equiaffine transformations. We then obtain explicit equations for all curves in 𝔻^2 whose equiaffine curvature is a dual constant. In particular, we prove that when the equiaffine curvature is a pure real constant, both the real and dual parts of the curve in 𝔻^2 are quadratic curves. In addition, we provide a complete classification of spacelike and timelike curves parametrized by a real variable in the dual Lorentz–Minkowski plane 𝔻^2_1 whose curvature is a dual constant.

We study pointwise hemislant submersions as a generalization of pointwise slant submersions and hemislant submersions from cosymplectic manifolds onto Riemannian manifolds. We investigate the integrability of distributions and the geometry of totally geodesic foliations arising from the definition of these submersions. Moreover, we study the ϕ-pluriharmonicity of these maps and obtain some inequalities connecting the Ricci curvature with the scalar curvature, depending on whether ξ is vertical or horizontal, for pointwise hemislant submersions from cosymplectic space forms onto Riemannian manifolds.

In this work, we move the study of generalized constant ratio hypersurfaces started in [6] into the Minkowski space. First, we get some geometrical properties of a non-degenerated GCR hypersurface in an arbitrary dimensional Minkowski space. Then, we obtain the complete classification of space-like GCR hypersurfaces with vanishing Gauss-Kronecker curvature in the Minkowski space E-1(n). We also give some explicit examples.

In the present paper, we discuss the singular minimal surfaces in a Euclidean 3-space R^{3} which are minimal. In fact, such a surface is nothing but a plane, a trivial outcome. However, a non-trivial outcome is obtained when we modify the usual condition of singular minimality by using a special semi-symmetric metric connection instead of the Levi-Civita connection on R^{3}. With this new connection, we prove that, besides planes, the singular minimal surfaces which are minimal are the generalized cylinders, providing their explicit equations. A trivial outcome is observed when we use a special semi-symmetric non-metric connection. Furthermore, our discussion is adapted to the Lorentz-Minkowski 3-space.

In this paper, we investigate the flow of curve and its equiform geometry in 4-dimensional Galilean space. We obtain that the Frenet equations and curvatures of inextensible flows of curves and its equiformly invariant vector fields and intrinsic quantities are independent of time. We find that the motions of curves and its equiform geometry can be defined by the inviscid and stochastic Burgers' equations in 4-dimensional Galilean space.

In this paper, we study and classify singular minimal translation surfaces in a Euclidean space of dimension $3$ endowed with a certain semi-symmetric (non-)metric connection.

In this paper, we consider the problem of finding the hypersurface M^n in the Euclidean (n+1)-space R^{n+1} that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the hypersurfaces in the upper halfspace (R^{n+1})_{+} with lowest gravity center, for a fixed unit vector u in R^{n+1} . We first state that a singular minimal cylinder M^n in R^{n+1} is either a hyperplane or a {\alpha}-catenary cylinder. It is also shown that this result remains true when M^n is a translation hypersurface and u a horizantal vector. As a further application, we prove that a singular minimal translation graph in R^3 of the form z=f(x)+g(y+cx), c in R-{0}, with respect to a certain horizantal vector u is either a plane or a {\alpha}-catenary cylinder.

In this paper, we study the problem of finding the affine factorable surfaces in a 3-dimensional isotropic space with prescribed Gaussian (K) and mean (H) curvature. Because the absolute figure two different types of these surfaces appear by permutation of coordinates. We firstly classify the affine factorable surfaces of type 1 with K,H constants. Afterwards, we provide the affine factorable surfaces of type 2 with K=const. and H=0. In addition, in a particular case, the affine factorable surfaces of type 2 with H=const. were obtained.

We introduce pointwise bi-slant submersions from cosymplectic manifolds onto Riemannian manifolds as a generalization of anti-invariant, semi-invariant, semi-slant, hemi-slant, pointwise semi-slant, pointwise hemi-slant and pointwise slant Riemannian submersions. We give an example for pointwise bi-slant submersions and investigate integrability and totally geodesicness of the distributions which are mentioned in the definition of pointwise bi-slant submersions admitting vertical Reeb vector field. Also we obtain necessary and sufficient conditions for such submersions to be totally geodesic maps.

In this paper, we study generalized constant ratio surfaces in the Euclidean 4-space. We also obtain a classifications of constant slope surfaces.

Bu calismada Oklidyen 3-uzayindaki Hasimoto yuzeyleri incelenmistir. Ilk olarak, Oklidyen 3-uzayindaki Hasimoto yuzeylerinin geometrik ozellikleri incelenmistir. Ozellikle Bishop catisi ile iliskilendirilmis bu yuzeylerin egrilikleri elde edilmistir. Daha sonrasinda bu yuzeylerin Bishop catisina gore parametre egrilerinin bazi karakterizasyonlari verilmistir.

In this paper, we characterize and classify helix surfaces with principal direction relatived to a space-like and light-like, constant direction in the Minkowski 3-space.

In this paper, we introduce canonical principal direction (CPD) submanifolds with higher codimension in Euclidean spaces. We obtain the complete classification of surfaces endowed with CPD in the Euclidean 4-space.

In this paper, we obtain the mean curvature of an A-net surface in the three-dimensional Heisenberg group H 3 . Moreover, we give some characterizations of this surface according to Levi-Civita connections of three dimensional Heisenberg group H 3 . Finally, we give an example and draw the minimal A-net surface with the help of Mathematica.

We define the e(1)(alpha) e(3)(alpha)-isotropic Smarandache curves of type-1 and type-2, the e(1)(alpha)e(2)(alpha)e(3)(alpha)-isotropic Smarandache curve, and the e(1)(alpha)e(2)(alpha)e(4)(alpha)-isotropic Smarandache curves of type-1 and type-2. Then we examine these kinds of isotropic Smarandache curve according to Cartan frame in the complex 4-space C-4 and give some differential geometric properties of these Samarandache curves.

In this paper, we study the position vector of a general helix according to type-2 Bishop frame in the 3-dimensional Euclidean space E3 . Moreover we determine the natural representation of a general helix in E^3 .

In this paper we study the minimal polynomial af ne translation surfaces in the 3-dimensional hyperbolic space H. We suppose that the af ne translation surface is minimal in E and then we prove that there are not any minimal polynomial af ne translation surfaces in H.

In this paper, we study the curves of constant breadth according to type-2 Bishop frame in the 3-dimensional Euclidean Space E-3. Moreover some characterizations of these curves are obtained.

In this paper, we characterize and classify all surfaces endowed with canonical principal direction relative to a space-like and light-like, constant direction in the Minkowski 3-space.

In this paper, we would like to give a short survey of recent results on hypersurfaces with canonical principal direction relative to a fixed direction in a (semi-)Riemannian manifold. We also present some of our first results that we have recently obtained in this direction.