Related papers: Bertrand Curves in three Dimensional Lie Groups
The aim of this paper is to study triharmonic curves in three dimensional f-Kenmotsu manifolds. We investigate necessary and sufficient conditions for Frenet curves, and specifically for slant and Legendre curves to be triharmonic. Then we…
In this paper, we consider the idea of Bertrand curves for curves lying on surfaces in Minkowski 3-space. By considering the Darboux frame, we define these curves as Bertrand D-curves and give the characterizations for those curves. We also…
In this paper, tangent-, principal normal-, and binormal-wise associated curves are defined such that each of these vectors of any given curve lies on the osculating, normal, and rectifying plane of its mate, respectively. For each…
In this study, we investigate Bertrand curves in three dimensional dual space D3 and we obtain the characterizations of these curves in dual space D3. Also we show that involutes of a curve constitute Bertrand pair curves.
The Frenet frame is generally known an orthonormal vector frame for curves. But, it does not always meet the needs of curve characterizations. In this study, with the help of associated curves of any spatial curve we obtained a new…
A new kind of partner curve called osculating mate of a Frenet curve is introduced. Some characterizations for osculating mate are obtained and using the obtained results some special curves such as slant helix, spherical helix, $C$-slant…
In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural…
In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the…
Membranes holomorphically embedded in flat noncompact space are constructed in terms of the degrees of freedom of an infinite collection of 0-branes. To each holomorphic curve we associate infinite-dimensional matrices which are static…
We define and study the harmonic curves on domains in $\mathbb{R}^n$ into the first Heisenberg group $\mathbb{H}^1$. These are the $C^2$-regular mappings which are critical points of the second Dirichlet energy and satisfy the weak…
The aim of this article is to characterize pairs of curves within multiplicative (non-Newtonian) spaces. Specifically, we investigate how famous curve pairs such as Bertrand partner curves, Mannheim partner curves, which are prominent in…
Perlick's classification of (3+1)-dimensional spherically symmetric and static spacetimes (\cal M,\eta=-1/V dt^2+g) for which the classical Bertrand theorem holds [Perlick V Class. Quantum Grav. 9 (1992) 1009] is revisited. For any Bertrand…
Bertrand's theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler…
We study the local equivalence problems of curves and surfaces in three dimensional Heisenberg group via Cartans method of moving frames and Lie groups, and find a complete set of invariants for curves and surfaces. For surfaces, in terms…
In this paper, we investigate a curve whose spherical image the tangent indicatrix and binormal indicatrix is slant helix and called it as a slant helix. We obtain that the spherical images are spherical slant helices defined by [3]. This…
The {\em focal curve} of an immersed smooth curve $\gamma:s\mapsto \gamma(s)$, in Euclidean space $\R^{m+1}$, consists of the centres of its osculating hyperspheres. The focal curve may be parametrised in terms of the Frenet frame of…
In this paper, we give a definition of harmonic curvature functions in terms of V_{n} and define a new kind of slant helix which is called V_{n}-slant helix in n-dimensional pseudo-Riemannian manifold. Also, we give important…
In this paper, we define slant helices in three dimensional Lie Groups with a bi-invariant metric and obtain a characterization of slant helices. Moreover, we give some relations between slant helices and their involutes, spherical images.
In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural…
In this study, some characterizations of Euler spirals in E_1^{3} have been presented by using their main property that their curvatures are linear. Moreover, discussing some properties of Bertrand curves and helices, the relationship…