Biharmonic Curves Research Articles

On p-biharmonic curves

In this article we study p-biharmonic curves as a natural generalization of biharmonic curves. In contrast to biharmonic curves p-biharmonic curves do not need to have constant geodesic curvature if p=12 in which case their equation reduces to the one of 12-elastic curves. We will classify 12-biharmonic curves on closed surfaces and three-dimensional space forms making use of the results obtained for 12-elastic curves from the literature. By making a connection to magnetic geodesic we are able to prove the existence of 12-biharmonic curves on closed surfaces. In addition, we will discuss the stability of p-biharmonic curves with respect to normal variations. Our analysis highlights some interesting relations between p-biharmonic and p-elastic curves.

Biharmonic curves along Riemannian maps

Biharmonic curves along Riemannian maps

An Intrinsic Version of the k-Harmonic Equation

The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle TkM to the cotangent bundle T*Tk−1M. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves.

A New Approach on Some Special Curves

In this paper, we obtain some characterizations for a Frenet curve with the help of an alternative frame different from Frenet frame. Also, in the present study we consider weak biharmonic and harmonic 1-type curves by using the mean curvature vector field of the curve. We also study 1-type and biharmonic curves whose mean curvature vector field is in the kernel of Laplacian. We give some theorems for them in the Euclidean 3-space. Moreover, we give the classifications of these type curves.

Generalized Cross Product in (2+s)-Dimensional Framed Metric Manifolds with Application to Legendre Curves

This study generalizes the cross product defined in 3-dimensional almost contact metric manifolds and describes a new generalized cross product for n=1 in (2n+s)-dimensional framed metric manifolds. Moreover, it studies some of the proposed product’s basic properties. It also performs characterizations of the curvature of a Legendre curve on an S-manifold and calculates the curvature of a Legendre curve. Furthermore, it shows that Legendre curves are also biharmonic curves. Next, this study observes that a Legendre curve of osculating order 5 on S-manifolds is imbedded in the 3-dimensional K-contact space. Lastly, the current paper discusses the need for further research.

Biharmonic Curves in Three-Dimensional Generalized Symmetric Spaces

In this paper, we study biharmonic curves in three-dimensio -nal generalized symmetric spaces, equipped with a left-invariant pseudo- Riemannian metric. We characterize non-geodesic biharmonic curves in three-dimensional generalized symmetric spaces and prove that there ex- ists no non-geodesic biharmonic spacelike helix in three-dimensional gen- eralized symmetric spaces. We also show that a linear map from a Eu- clidean space in three-dimensional generalized symmetric spaces is bihar- monic if and only if it is a harmonic map, and give a complete classification of such maps.

Biharmonic Curves in a Strict Walker 3-Manifold

In this paper, we study the geometry of biharmonic curves in a strict Walker 3-manifold and we obtain explicit parametric equations for biharmonic curves and time-like biharmonic curves, respectively. We discuss the conditions for a speed curve to be a slant helix in a Walker manifold. We give an example of biharmonic curve for illustrating the main result.

Biharmonic curves in f-Kenmotsu 3-manifolds

It is known that there exist no proper biharmonic helices in Kenmotsu 3-manifolds. In this paper we show the existence of proper biharmonic helices in certain f-Kenmotsu 3-manifolds.

Biharmonic curves in $3$-dimensional Lorentzian Sasakian space forms

Biharmonic curves in $3$-dimensional Lorentzian Sasakian space forms

Totally biharmonic hypersurfaces in space forms and 3-dimensional BCV spaces

A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi–Cartan–Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional sphere, the only nontrivial example of a totally biharmonic surface appears in the product space [Formula: see text].

SOME SPECIAL CURVES IN THREE DIMENSIONAL f-KENMOTSU MANIFOLDS

In this paper we study Biharmonic curves, Legendre curves and Magnetic curves in three dimensional f-Kenmotsu manifolds. We also study 1-type curves in a three dimensional f-Kenmotsu manifold by using the mean curvature vector field of the curve. As a consequence we obtain for a biharmonic helix in a three dimensional f-Kenmotsu manifold with the curvature κ and the torsion τ, κ2+τ2 = -(f2 +f′). Also we prove that if a 1-type non-geodesic biharmonic curve γ is helix, then λ= -(f2 + f′):

F−BIHARMONIC CURVES WITH TIMELIKE NORMAL VECTOR ON LORENTZIAN SPHERE

In this paper, we study $f-$biharmonic curves as the critical points of the $f-$bienergy functional $E_{2}(\psi )=\int_{M}f\mid \tau (\psi )^{2}\mid \vartheta _{g}$, on a Lorentzian para-Sasakian manifold $M$. We give necessary and sufficient conditions for a curve such that has a timelike principal normal vector on lying a $4$-dimensional conformally flat, quasi-conformally flat and conformally symmetric Lorentzian para-Sasakian manifold to be an $f-$biharmonic curve. Moreover, we introduce proper $f-$biharmonic curves on the Lorentzian sphere $S_{1}^{4}.$

Certain Curves Associated with f -Kenmotsu Manifolds

The aim of the present paper is to study slant magnetic curves, magnetic curves and biharmonic curves on a 3-dimensional f -Kenmotsu mani- fold. Next we deal with curve for which tangent vector of the curve is parallel to ξ. Also we characterize ξ-vertical hypersurface in a 3-dimensional f -Kenmotsu manifold.

On Pseudo-Hermitian Biharmonic Slant Curves in Sasakian Space Forms Endowed with the Tanaka–Webster Connection

In this paper, we consider pseudo-Hermitian biharmonic slant curves in $$(2n+1)$$-dimensional Sasakian manifolds endowed with the Tanaka–Webster connection. We investigate pseudo-Hermitian curvatures of slant curves in six different cases by solving differential equations and linear independency of Frenet frame vector fields. We also construct an example of pseudo-Hermitian slant curves in $${\mathbb {R}}^{7}(-3)$$.

A Note on Quasi Biharmonic Curves According to Quasi Frame in the Minkowski Space

A Note on Quasi Biharmonic Curves According to Quasi Frame in the Minkowski Space

A New Version of Quasi Biharmonic Curves with Quasi Frame

A New Version of Quasi Biharmonic Curves with Quasi Frame

Inextensible flows of biharmonic S-curves according to Sabban frame in Heisenberg group Heis3

In this work, we examine inextensible flows associated biharmonic curves depending to Sabban frame in Heis3. We define the biharmonic curves with regards to their curvatures. The differential equations characterizing biharmonic curves are given in Heis3. Additionaly, we discover equations of one parameter family of biharmonic curves in .

General helices with lightlike slope axis

In this paper, we investigate general helices with lightlike slope axis. We give necessary and sufficient conditions for a general helix to have a lightlike slope axis. We obtain parametric equation of all general helices with lightlike slope axis. Also we give a nice relation between helix with lightlike slope axis and biharmonic curves in Minkowski 3-space E31.

CHARACTERIZATIONS OF SPACE CURVES WITH 1-TYPE DARBOUX INSTANTANEOUS ROTATION VECTOR

Abstract. In this study, by using Laplace and normal Laplace operators,we give some characterizations for the Darboux instantaneous rotationvector field of the curves in the Euclidean 3-space E 3 . Further, we givenecessary and sufficient conditions for unit speed space curves to have1-type Darboux vectors. Moreover, we obtain some characterizations ofhelices according to Darboux vector. 1. IntroductionOne of the most important problems of local differential geometry is toobtain the relations characterizingspecial curves with respect to their curvatureand torsion. The well-known types of such special curves are constant slopecurves or general helices which are defined by the property that the tangentvectors of curves make a constant angle with fixed directions. A necessaryand sufficient condition for a curve to be a general helix in the Euclidean 3-space E 3 is that the ratio of curvature to torsion is constant [11]. So, manymathematicians have focused their studies on these special curves in differentspaces such as Euclidean space and Minkowski space [3, 4, 5, 10].Furthermore, Chen and Ishikawa [1] classified biharmonic curves, the curvesfor which ∆H~ = 0 holds in semi-Euclidean space E

Mannheim partner curves in Cartan-Vranceanu 3-space

In this paper, the Mannheim mate curves of the proper biharmonic curves in Cartan-Vranceanu 3-dimensional spaces (M,ds2l,m), with l2 ? 4m and m ? 0 are studied. We give the definition of the Mannheim mate of a proper biharmonic curve and give the explicit parametric equations of that Mannheim mate curve in Cartan-Vranceanu 3-dimensional space. Moreover, we show that the distance between corresponding points of the Mannheim pairs is constant in Cartan-Vranceanu 3-dimensional spaces.