A Novel Condition to the Harmonic of the Velocity Vector Field of a Curve in R n

Differential geometry is applied to other fields of science and mathematics. In particular, it applied various problems in mechanics, computer-aided as well as traditional engineering design, physics, geodesy, geography, space travel, and relativity theory [1]. The volume of unit vector fields has been studied byGluck and Ziller [2], Johnson [3], and Higuchi et al. [4], among other scientists. In [5], the energy of a unit vector field X on a Riemannian manifoldM is defined as the energy of the mappingX : M → T1M, where the unit tangent bundleT1M is equipped with the restriction of the Sasaki metric on TM. Generally, any geometric problem about curves can be solved using the curves’ Frenet vectors field.Therefore, in [6], we focus on the curve C instead of the manifold M. For a given curve C, with a pair of parametric unit speeds (I, α) in a space Rn we denote Frenet frames at the points α(a) and α(s) by {V1(a), V2(a), . . . , Vr(a)} and {V1(s), V2(s), . . . , Vr(s)}, respectively, as we take a fixed point a ∈ I. We calculate the energy of the Frenet vectors fields and the angle between the vectors Vi(a) and Vi(s), where 1 ≤ i ≤ r. So, we see that both energy and angle depend on the curvature functions of the curve C. In this paper, we choose two points p and q in Rn. We obtain a condition for the harmonic of the velocity vector field in the curve family of all curves from p to q points. Thus, we notice that this condition can be expressed in terms of the curvature functions. Finally, we give an example which provides the mentioned condition in this work and illustrate it with figures.


Introduction
Differential geometry is applied to other fields of science and mathematics.In particular, it applied various problems in mechanics, computer-aided as well as traditional engineering design, physics, geodesy, geography, space travel, and relativity theory [1].
The volume of unit vector fields has been studied by Gluck and Ziller [2], Johnson [3], and Higuchi et al. [4], among other scientists.In [5], the energy of a unit vector field  on a Riemannian manifold  is defined as the energy of the mapping  :  →  1 , where the unit tangent bundle  1  is equipped with the restriction of the Sasaki metric on .
Generally, any geometric problem about curves can be solved using the curves' Frenet vectors field.Therefore, in [6], we focus on the curve  instead of the manifold .For a given curve , with a pair of parametric unit speeds (, ) in a space   we denote Frenet frames at the points () and () by { 1 (),  2 (), . . .,   ()} and { 1 (),  2 (), . . .,   ()}, respectively, as we take a fixed point  ∈ .We calculate the energy of the Frenet vectors fields and the angle between the vectors   () and   (), where 1 ≤  ≤ .So, we see that both energy and angle depend on the curvature functions of the curve .
In this paper, we choose two points  and  in   .We obtain a condition for the harmonic of the velocity vector field in the curve family of all curves from  to  points.Thus, we notice that this condition can be expressed in terms of the curvature functions.Finally, we give an example which provides the mentioned condition in this work and illustrate it with figures.
Definition 2. Let (, ) be a parametric pair for a curve  in a space   and { 1 (),  2 (), . . .,   ()} be Frenet frames at the point () ∈ .Let be defined as curvature function on  and the real number   () be defined as th curvature on  at the point ().
This gives a Riemannian metric on .Recall that   is called the Sasaki metric.The metric   makes the projection  :  1  →  a Riemannian submersion [8].
Definition 6.The energy of a differentiable map  : (, ) → (, ℎ) between Riemannian manifolds is given by where V  is the canonical volume form in  and {  } is a local basis of the tangent space [5,9].
Let  ∞ (; ) denote the space of all smooth maps from  to .A map  :  →  is said to be harmonic if it is an extremal (i.e., critical point) of the energy functional E(⋅; ) :  ∞ (; ) →  for any compact domain .

A Condition for the Curve Where the Velocity Vector Field Is Harmonic
The following theorem characterizes a critical point of the energy of the velocity vector field of a curve in   .
Theorem 8. Let  be unit speed curve in   and () = , () = .If the velocity vector field of  along from  to  is harmonic, then the following equation is satisfied: where  1 is the 1th curvature function and  is the real-valued function on [, ].

Figures 1 and 2
Figures 1 and 2 are shown in 3-dimensional space as sample to -dimensional space.