Abstract
In this paper, a condition is obtained for the harmonic of the velocity vector field in the curve family passing through the fixed and points in . It shows that the condition can be expressed in terms of the curvature functions. Finally, we give an example which provides the mentioned condition in this work and illustrates it with figures.
1. 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 , where the unit tangent bundle 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 and , respectively, as we take a fixed point . We calculate the energy of the Frenet vectors fields and the angle between the vectors and , where 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 1. A curve segment is the portion of a curve defined in a closed interval [7].
Definition 2. Let be a parametric pair for a curve in a space and be Frenet frames at the point . Letbe defined as curvature function on and the real number be defined as th curvature on at the point .
Theorem 3 (Frenet formulas). Let be a parametric pair for a curve in a space . If we take th curvature and Frenet frames at the point , then the following relations are hold:
Proposition 4. The connection map verifies the following conditions:(1) and , where is the tangent bundle projection and is the bundle projection.(2)For and a section , we have where is the Levi-Civita covariant derivative [8].
Definition 5. For we defineThis gives a Riemannian metric on . Recall that is called the Sasaki metric. The metric makes the projection a Riemannian submersion [8].
Definition 6. The energy of a differentiable map between Riemannian manifolds is given bywhere 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 for any compact domain .
By a (smooth) variation of we mean a smooth map such that . We can think of as a family of smooth mappings which depends “smoothly” on a parameter .
Definition 7. In [10], A smooth map is said to be harmonic if for all compact domains and all smooth variations of supported in , where
2. 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 α be unit speed curve in and , . If the velocity vector field of along from to is harmonic, then the following equation is satisfied:where is the 1th curvature function and λ is the real-valued function on .
Proof. Let be a unit speed curve in and , , . There exists a real-valued function on , , , and for all . Let be the Frenet frame field on andLet the collection of curves befor sufficiently small .
For , , and , we have and ,
These results show that is the curve segment from to . Assume this collection for all curves. The expression for the energy of the vector field of from to becomes .
Now, let be the tangent bundle. So we have , where , , and denotes straight line generated . Let be the bundle projection. By using (5) we calculate the energy of as where is the differential arc length. From (4) we haveSince is a section, we have By Proposition 4, we also have thatgivingUsing these results in (10) we get where ; .
By Definition 7, if is a harmonic, then should be a critical point of . Suppose that .
From (14) we obtainSince we have and we getWe can writeThus, we can deduceSubstituting (18) in (16), for ,From (8) and (9), we obtainNow we calculate the partial derivatives of (22) with respect to and ; using Frenet formulas, we getFrom (21), we haveIt follows from (23) and (25) thatConsidering the candidate function , we getFrom (23), we getTherefore, (25) and (28) give Substituting (27) and (29) in (20) yieldsSince , it gives andThis completes the proof of the theorem. Also, it is trivial that geodesics and curves with constant curvature satisfy the theorem.
We give an example that provides the condition (7) in the theorem below. Using different curvatures, we illustrate the example with Figures 1 and 2.
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(b)
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Example 9. Let , , , . If we can choose , , , , and for all then the curves which have , ( and ) provide condition (7).
Figures 1 and 2 are shown in 3-dimensional space as sample to -dimensional space.
Conflicts of Interest
The authors declare that they have no conflicts of interest.