# A Novel Condition to the Harmonic of the Velocity Vector Field of a Curve inRn

Authors: Ayşe Altın; Canan Köroğlu
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

In this paper, a condition is obtained for the harmonic of the velocity vector field in the curve family passing through the fixedp and q points in Rn. 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.


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## Body

## 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 X on a Riemannian manifold M is defined as the energy of the mapping X:M→T1M, where the unit tangent bundle T1M 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 V1a,V2a,…,Vr(a) and V1s,V2s,…,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 Via and Vis, 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 pointsp 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.Definition 1.
A curve segment is the portion of a curve defined in a closed interval[a,b] [7].Definition 2.
Let(I,α) be a parametric pair for a curve C in a space Rn and V1(s),V2(s),…,Vr(s) be Frenet frames at the point αs∈C. Let(1)kis=Vi′s,Vi+1s,∀s∈I,1≤i≤rbe defined as curvature function on C and the real number kis be defined as ith curvature on C at the point α(s).Theorem 3 (Frenet formulas).
Let(I,α) be a parametric pair for a curve C in a space Rn. If we take ith curvature kis and Frenet frames V1s,V2s,…,Vrs at the point αs, then the following relations are hold:(2)Vi′s=k1sV2s,Vi′s=-ki-1sVi-1s+kisVi+1s,1≤i≤rVr′s=-kr-1sVr-1s.Proposition 4.
The connection mapK:T(T1M)→T1M verifies the following conditions:(1)
π∘K=π∘dπ and π∘K=π∘π~, where π~:T(T1M)→T1M is the tangent bundle projection and π:T1M→M is the bundle projection.(2)
Forω∈TxM and a section ξ:M→T1M, we have(3)Kdξω=∇ωξ,where∇ is the Levi-Civita covariant derivative [8].Definition 5.
Forη1,η1∈Tξ(T1M) we define(4)gsη1,η1=dπη1,dπη2+Kη1,Kη2.This gives a Riemannian metric on TM. Recall that gs is called the Sasaki metric. The metric gs makes the projection π:T1M→M a Riemannian submersion [8].Definition 6.
The energy of a differentiable mapf:(M,g)→(N,h) between Riemannian manifolds is given by(5)Ef=12∫M∑a=1nhdfea,dfeavg,where vg is the canonical volume form in M and ea is a local basis of the tangent space [5, 9].LetC∞(M;N) denote the space of all smooth maps from M to N. A map f:M→N is said to be harmonic if it is an extremal (i.e., critical point) of the energy functional E·;D:C∞(M;N)→R for any compact domain D.By a (smooth) variation off we mean a smooth map :M×-ε,ε→N,x,t→ftx(ε>0) such that f0=f. We can think of {ft} as a family of smooth mappings which depends “smoothly” on a parameter t∈-ε,ε.Definition 7.
In [10], A smooth map f:(M,g)→(N,h) is said to be harmonic if (6)ddtEft;Dt=0=0for all compact domains D and all smooth variations {ft} of f supported in D, where Ef;D=1/2∫D∑a=1nhdfea,df(ea)vg.


## 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 inRn.Theorem 8.
Letα be unit speed curve in Rn and α(a)=p, α(b)=q. If the velocity vector field of α along from p to q is harmonic, then the following equation is satisfied:(7)∫abλsk1sk1′sds=0,where k1 is the1th curvature function and λ is the real-valued function on [a,b].Proof.
Letα:I→Rn be a unit speed curve in Rn and [a,b]⊂I, α(a)=p, α(b)=q. There exists a real-valued function λ on [a,b], λ(s)=(s-a)(b-s), λ(a)=λ(b)=0, and λ(s)≠0 for all s∈(a,b). Let {V1,V2,…,Vr} be the Frenet frame field on α and(8)λsV1s=v1s,v2s,…,vns,vi:a,b⟶R.Let the collection of curves be(9)αks=α1s+kv1s,α2s+kv2s,…,αns+kvnsfor sufficiently small k.
Fork=0, α0(s)=α(s), and λ(a)=λ(b)=0, we have vi(a)=vi(b)=01≤i≤n and αka=p, αkb=q.
These results show thatαk is the curve segment from p to q. Assume this collection αks=α(s,k) for all curves. The expression for the energy of the vector field V1k of αk from p to q becomes E(V1k).
Now, letTCk be the tangent bundle. So we have V1k:Ck→TCk, where TCk=∪t∈ITαk(t)Ck, Ck=αkI, and Tαk(t)Ck denotes straight line generated V1k. Let π:TCk→Ck be the bundle projection. By using (5) we calculate the energy of V1k as (10)EV1k=12∫abgSdV1kV1kαs,k,dV1kV1kαs,kds,where ds is the differential arc length. From (4) we have(11)gSdV1kV1k,dV1kV1k=dπdV1kV1k,dπdV1kV1k+KdV1kV1k,KdV1kV1k.Since V1k is a section, we have d(π)∘d(V1k)=dπ∘V1k=didCk=idTCk. By Proposition 4, we also have that(12)KdV1kV1k=∇V1kV1k=V1′k=∂V1k∂s,giving(13)gSdV1kV1k,dV1kV1k=V1k,V1k+V1′k,V1′k.Using these results in (10) we get (14)EV1k=12∫abV1k,V1k+V1′k,V1′kds,where V1k=1/ws,kdα/∂s(s,k); ws,k=dα/∂s(s,k),dα/∂s(s,k).
By Definition7, if V1k is a harmonic, then k=0 should be a critical point of EV1k. Suppose that ∂EV1k/∂kk=0=0.
From (14) we obtain(15)∂EV1k∂k=∂∂k12∫abV1k,V1k+V1′k,V1′kds=12∫ab∂∂kV1k,V1k+∂V1k∂s,∂V1k∂sds.Since V1k,V1k=1 we have ∂/∂kV1k,V1k=0 and we get(16)∂EV1k∂k=12∫ab∂∂k∂V1k∂s,∂V1k∂sds=∫ab∂2V1k∂s∂k,∂V1k∂sds.We can write(17)∂∂s∂V1k∂k,∂V1k∂s=∂2V1k∂s∂k,∂V1k∂s+∂V1k∂k,∂2V1k∂s2.Thus, we can deduce(18)∂2V1k∂s∂k,∂V1k∂s=∂∂s∂V1k∂k,∂V1k∂s-∂V1k∂k,∂2V1k∂s2.Substituting (18) in (16), for k=0,(19)∂EV1k∂kk=0=∫ab∂∂s∂V1k∂ks,0,∂V1k∂ss,0-∂V1k∂ks,0,∂2V1k∂s2s,0ds,(20)∂EV1k∂kk=0=∂V1k∂ks,0,∂V1k∂ss,0ab-∫ab∂V1k∂ks,0,∂2V1k∂s2s,0ds.From (8) and (9), we obtain(21)dα∂ks,k=λsV1s,(22)dα∂ss,0=α′s=V1s,0.Now we calculate the partial derivatives of (22) with respect to s and k; using Frenet formulas, we get(23)∂V1k∂ss,0=∂2α∂s2s,0=α′′s=V1′k=k1sV2s,(24)∂V1k∂ks,k=∂2α∂s∂ks,k=∂2α∂k∂ss,k.From (21), we have(25)∂V1k∂ks,kk=0=∂V1k∂ks,0=λ′sV1s+λsk1sV2s.It follows from (23) and (25) that(26)∂V1k∂ks,0,∂V1k∂ss,0=λsk12s.Considering the candidate function λa=λb=0, we get(27)∂V1k∂ks,0,∂V1k∂ss,0ab=λbk12b-λak12a=0.From (23), we get(28)∂2V1k∂s2s,0=-k12sV1s+k1′sV2s+k1sk2sV3s.Therefore, (25) and (28) give (29)∂V1k∂ks,0,∂2V1k∂s2s,0=-λsk12s′+3λsk1sk′1s.Substituting (27) and (29) in (20) yields(30)∂EV1k∂kk=0=-∫ab-λsk12s′+3λsk1sk′1sds,∂EV1k∂kk=0=-λsk12sab-3∫abλsk1sk′1sds=0.Since λa=λb=0, it gives -λsk12sab=0 and(31)∂EV1k∂kk=0=-3∫abλsk1sk′1sds=0.This 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.Figure 1
(a)κ1s=5s2+3,κ2(s)=0 (b) κ1s=s2+1,κ2(s)=0.
(a)
(b)Figure 2
(a)κ1s=5s,κ2(s)=0, (b) κ1s=5s5,κ2(s)=0.
(a)
(b)Example 9.
Letα:I→R3, [-1,1]⊂I, α(-1)=p, α(1)=q. If we can choose λ:[-1,1]→R, λ(s)=1-s2, λ(-1)=0, λ(1)=0, and λ(s)≠0 for all s∈(-1,1) then the curves which have κs=cs2n+d, κs=cs2n+1 (c,d∈R and n∈N) provide condition (7).Figures1 and 2 are shown in 3-dimensional space as sample to n-dimensional space.



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*Source: 7142412-2017-06-08.xml*