Related papers: Smarandache Curves According to Bishop Frame in Eu…
Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.
In this study, we investigate the locus of the centers of the Meusnier spheres. Just as focal curve is the locus of the centers of the osculating spheres, we investigate the geometrical interpretation on the locus of the centers of the…
In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized spherical curves in Euclidean $(n+1)-$space $\mathbb{E}^{n+1}$. Further, we introduce some kind of generalized…
The evolute of a smooth curve in an m-dimensional Euclidean space is the locus of centers of its osculating spheres, and the evolute of a spatial polygon is the polygon whose consecutive vertices are the centers of the spheres through the…
The well known formulas express the curvature and the torsion of a curve in $R^3$ in terms of euclidean invariants of its derivatives. We obtain expressions of this kind for all curvatures of curves in $R^n$. It follows that a curve in…
We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmu\c{s}-Montaldo-Oniciuc, we…
In this paper we consider the complete biconservative surfaces in Euclidean space $\mathbb{R}^3$ and in the unit Euclidean sphere $\mathbb{S}^3$. Biconservative surfaces in 3-dimensional space forms are characterized by the fact that the…
An analytic solution for Helfrich spontaneous curvature membrane model (H. Naito, M.Okuda and Ou-Yang Zhong-Can, Phys. Rev. E {\bf 48}, 2304 (1993); {\bf 54}, 2816 (1996)), which has a conspicuous feature of representing the circular…
The paper is devoted to differential geometric invariants determining a Frenet curve in up to a direct similarity These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then,…
This paper is a continuation of the papers [2,3,4,5,6]. In this paper the osculating spaces of arbitrary order of a manifold embedded in Euclidean space are considered. A better estimation of their dimensions as well as the description of…
In this paper, we investigate constant breadth curves on a surface according to Darboux frame and give some characterizations of these curves.
In this paper, we investigate special curves on a strong r-helix submanifold in Euclidean n-space E n. Also, we give the important relations between strong r-helix submanifolds and the special curves such as line of curvature, geodesic and…
A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are…
In this paper, helicoidal flat surfaces in the $3$-dimensional sphere $\mathbb{S}^3$ are considered. A complete classification of such surfaces is given in terms of their first and second fundamental forms and by linear solutions of the…
In this paper, we give smoe characterizations of relatively normal-slant helices and isophotic curves on a smooth surface immersed in Euclidean 3-space with respect to their position vevtor. We also introduce the methods for generating an…
The orthogonal trajectories of the first tangents of the curve are called the involutes of $x$. The hyperspheres which have higher order contact with a curve $x$ are known osculating hyperspheres of $x$. The centers of osculating…
We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms.
In this study, we consider canal surfaces according to parallel transport frame in Euclidean space $\mathbb{E}^{4}$. The curvature properties of these surfaces are investigated with respect to $k_{1}$, $k_{2}$ and $k_{3}$ which are…
We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an…
A Smarandache multi-space is a union of $n$ different spaces equipped with some different structures for an integer $n\geq 2$, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in…