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Volume 14, Issue 1 (Jan 2016)


On new characterization of inextensible flows of space-like curves in de Sitter space

Mustafa Yeneroğlu
  • Corresponding author
  • Firat University, Department of Mathematics, 23119, Elaziğ, Turkey
  • Email:
Published Online: 2016-11-25 | DOI: https://doi.org/10.1515/math-2016-0071


Elastica and inextensible flows of curves play an important role in practical applications. In this paper, we construct a new characterization of inextensible flows by using elastica in space. The inextensible flow is completely determined for any space-like curve in de Sitter space S13. Finally, we give some characterizations for curvatures of a space-like curve in de Sitter space S13.

Keywords: Fluid flow; Minkowski space-time; Partial differential equation; de Sitter space 𝕊31

MSC 2010: 53A04; 53A05

1 Introduction

In mathematics and physics, a de Sitter space is the analogue in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space, denoted dSn, is the Lorentzian manifold analogue of an n-sphere (with its canonical Riemannian metric). It is maximally symmetric, has constant positive curvature, and is simply connected for n at least 3. More recently, it has been considered as the setting for special relativity rather than using Minkowski space, since a group contraction reduces the isometry group of de Sitter space to the Poincaré group, allowing a unification of the spacetime translation subgroup and Lorentz transformation subgroup of the Poincaré group into a simple group rather than a semi-simple group. This alternate formulation of special relativity is called de Sitter relativity, [16].

The elastica caught the attention of many of the brightest minds in the history of mathematics, including Galileo, the Bernoullis, Euler, and others. It was present at the birth of many important fields, most notably the theory of elasticity, the calculus of variations, and the theory of elliptic integrals. The path traced by this curve illuminates a wide range of mathematical style, from the mechanics-based intuition of the early work, through a period of technical virtuosity in mathematical technique, to the present day where computational techniques dominate [1620].

The flow of a curve or surface is said to be inextensible if, in the former case, the arc-length is preserved, and in the latter case, if the intrinsic curvature is preserved [7]. Physically, inextensible curve and surface flows are characterized by the absence of any strain energy induced from the motion. In [15], Kwon investigated inextensible flows of curves and developable surfaces in R3. Necessary and sufficient conditions for an inextensible curve flow were first expressed as a partial differential equation involving the curvature and torsion. Then, they derived the corresponding equations for the inextensible flow of a developable surface, and showed that it suffices to describe its evolution in terms of two inextensible curve flows, [15]. Flows of curves of a given curve are also widely studied, [814].

This study is organised as follows: Firstly, we construct a new method for inextensible flows of space-like curves in de Sitter space S13. Secondly, using the Frenet frame of the given curve, we present partial differential equations. Finally, we give some characterizations for curvatures of a curve in de Sitter space S13.

2 New geometry of space-like curves in S13-space

It is well-known that the Lorentzian space form with a positive curvature, more precisely [1], a positive sectional curvature is called de Sitter space S13. We define de Sitter 3-space by S13={xR14|x,x=1}.

It is well-known that to each unit speed space-like curve γ:IS13 one can associate a pseudo-orthonormal frame{γ, T, N, B}. Denote by {T, N, B} the space-like tangent vector, the space-like principal normal vector, and the time-like binomial vector, respectively. In this situation, the Frenet-Serret equations satisfied by the Frenet vectors {T, N, B} formally given by γ=T,TT=γ+κN,TN=κδ(γ)T+τB,TB=τN,(1)

where δ(γ) = –sign(N), and κ, τ are the curvature and the torsion of a curve γ respectively and given by κ=T+γ,τ(s)=δ(γ)R2det(γ,γ,γ,γ),

with R(s) ≠ 0.

Let γ(u, w) be a one parameter family of smooth space-like curves in S13. γw=π1T+π2N+π3B

Putting W=W(w,t)=γw,V(u,w)=γu=v(u,w)T(u,w),

which gives [W,T]=W(v)vT=gT.

Finally [18], we obtain that W(v)=gv,g=<TW,T>.

Definition 2.1: The flow γw in de Sitter space S13 is said to be inextensible if wαu=0.(2)

Theorem 2.2: Let γw. be a smooth flow of γ. The flow is inextensible if and only if π2vκδ(γ)+π1u=0(3)Now, assume that γ is arc-length parametrized curve. Then, we have

Lemma 2.3: wT=π1γ+[π1κ+π3τ+π2s]N+[π2τ+π3s]B, where π1, π2, π3 are smooth functions of time and arc-length.

Proof: From the definition of inextensible flow, we have wT=π1γ+[π2κδ(γ)+π1s]T+[π1κ+π3τ+π2s]N+[π2τ+π3s]B.Using Eq. (3), we obtain Eq. (4). This completes the proof.Now we give the characterization of evolution of first curvature as below:

Theorem 2.4: Let γ be one parameterfamily curves in de Sitter space S13.ifγω is inextensible flow of space-like γ in de Sitter space S13, then the evolution of κ is given by κw=s[π1κ+π3τ+π2s]+τ[π2T+π3s]+1κπ1s+π2, where π1, π2, π3 are smooth functions of time and arc-length.

Proof: A differentiation in Eq. (4) and the Frenet formulas give us that swT=π1sγ[π1[π1κ+π3τ+π2s]κδ(γ)]T+[s[π1κ+π3τ+π2s]+τ[π2τ+π3s]]N+[s[π2T+π3s]+τ[π1κ+π3τ+π2s]]B.Using the formula of the curvature, we write a relation wsTswT=R(γs,γw)T.We immediately arrive at R(γs,γw)T=π2R(T,N)T+π3R(T,B)T.Another important fact is that the curvature operator R on de Sitter space S13 has a simple expression, i.e., R(X1,X2)X3=g(X1,X3)X2g(X2,X3)X1.Then, R(T,N)T=g(T,T)Ng(N,T)T=N,R(T,B)T=g(T,T)Bg(B,T)T=B.From above equations, we get R(γs,γw)T=π2N+π3B.Then, we can write wsT=swT+R(γs,γw)T.Thus it is easy to obtain that wsT=π1sγ[π1[π1κ+π3τ+π2s]κδ(γ)]T +[s[π1κ+π3τ+π2s]+τ[π2τ+π3s]+π2]N+[s[π2τ+π3s]+τ[π1κ+π3τ+π2s]+π3]B.On the other hand, we have κw=wg(sT,N).Since, we express κw=g(wsT,N)+g(sT,wN).Moreover, by the definition of metric tensor, we have g(N,wN)=0.Then g(sT,wN)=g(γ,wN)=1κπ1s.Combining these we have κw=s[π1κ+π3τ+π2s]+τ[π2κ+π3s]+1κπ1s+π2.Thus, we obtain the theorem. This completes the proof.From the above theorem, we have

Theorem 2.5: wN=1κπ1sγ+1κ[[π1κ+π3τ+π2s]κδ(γ)]T+1κ[s[π2τ+π3s]+τ[π1κ+π3τ+π2s]+2π3]B, where π1, π2, π3 are smooth functions of time and arc-length.

Proof: Using Frenet equations, we have wsT=γw+κwN+κwN.Then, κwN=π1sγ+[[π1κ+π3τ+π2s]κδ(γ)]T+[κt+s[π1κ+π3τ+π2s]+τ[π2τ+π3s]+2π2]N+[s[π2τ+π3s]+τ[π1κ+π3τ+π2s]+2π3]B.Therefore g(N,wN)=0.From above equation we obtain wN=1κπ1sγ+1κ[[π1κ+π3τ+π2s]κδ(γ)]T+1κ[κt+s[π1κ+π3τ+π2s]+τ[π2τ+π3s]+2π2]N+1κ[s[π2τ+π3s]+τ[π1κ+π3τ+π2s]+2π3]B, which completes the proof.

Theorem 2.6: Let γ be one parameterfamily curves in de Sitter space S13.Ifγw is inextensible flow of space-like γ in de Sitter space S13 then wB=1τ[π1κδ(γ)+s[1κπ1s][1κ[[π1κ+π3τ+π2s]κδ(γ)]]]γ+1τ[π2+s[1κ[[π1κ+π3τ+π2s]κδ(γ)]]1κπ1stκδ(γ)]T+1τ[κ[1κ[[π1κ+π3τ+π2s]κδ(γ)]]+τ[1κ[s[π2T+π3s]+τ[π1κ+π3τ+π2s]+2π3]]κδ(γ)[π1κ+π3τ+π2s]]N, where π1, π2, π3 are smooth functions of time and arc-length.

Proof: Assume that γw be inextensible flow of γ.swN=[s[1κπ1s][1κ[[π1κ+π3τ+π2s]κδ(γ)]]]γ+[s[1κ[[π1κ+π3τ+π2s]κδ(γ)]]1κπ1s]T+[κ[1κ[[π1κ+π3τ+π2s]κδ(γ)]]+τ[1κ[s[π2τ+π3s]+τ[π1κ+π3τ+π2s]+2π3]]]N+s[1κ[s[π2τ+π3s]+τ[π1κ+π3τ+π2s]+2π3]]B.Under the assumption of space-like curve, we have wsN=π1κδ(γ)γ+tκδ(γ)T+κδ(γ)[π1κ+π3τ+π2s]N+[κδ(γ)[π2τ+π3s]+τt]B+τwB.Using the formula of the curvature, we write a relation wsNswN=R(γs,γw)N.Thus, it is seen that R(γs,γw)N=π2R(T,N)N+π3R(T,B)N.By using formula of curvature, we have R(T,N)N=g(T,N)Ng(N,N)T=T,R(T,B)N=g(T,N)Bg(B,N)T=0.Arranging the last equations, we obtain R(γs,γw)N=π2T.Therefore, we can easily see that τwB=[π1κδ(γ)+s[1κπ1s][1κ[[π1κ+π3τ+π2s]κδ(γ)]]]γ+[π2+s[1κ[[π1κ+π3τ+π2s]κδ(γ)]]1κπ1stκδ(γ)]T+[κ[1κ[[π1κ+π3τ+π2s]κδ(γ)]]+τ[1κ[s[π2τ+π3s]+τ[π1κ+π3τ+π2s]+2π3]]κδ(γ)[π1κ+π3τ+π2s]]N +[s[1κ[s[π2T+π3s]+τ[π1K+π3τ+π2s]+2π3]][κδ(γ)[π2T+π3s]+τt]]B.By this way, we conclude g(B1,ωB1)=0.Thus, we obtain the theorem. The proof of theorem is completed. Now we give the characterization of evolution of second curvature as below:

Theorem 2.7: Let γ be one parameterfamily curves in de Sitter space S13.ifγt is inextensible flow of space-like γ in de Sitter space S13 then the evolution of τ is given by τw=s[1κ[s[π2T+π3s]+τ[π1K+π3τ+π2s]+2π3]]κδ(γ)[π2T+π3s],where π1, π2, π3, are smooth functions of time and arc-length.

proof: It is obvious from Theorem 2.6. This completes the proof.Since δ(t) is an immersed curve, it has velocity vector V=vT and squared geodesic curvature κ2+1=TT2.

Theorem 2.8 (Main Theorem): W(κ2+1)=2<ssW,sT>+4g(κ2+1)+2<R(W,T)T,sT>,where g=<TW,T>.

proof: From Euler equations, we easily have W(κ2+1)=2<ssW,sT>+2<R(W,T)T,sT>+4g<sT,sT>.

Corollary 2.9: W(κ2+1)=W(κ2)In what follows, γ: [0,1] → M is a curve of length L. Now for fixed constant λ let Fλ(γ)=120Lκ2+1+λds=1201(sT2+λ)v(t)dt.For a variation γω with variation field M, we compute ddwFλ(γw)=1201W(κ2+1)v+(κ2+1+λ)W(v)dt=1201W(κ2+1)(κ2+1+λ)gds=01<ssW,sT<+2g(κ2+1)+>R(W,T)T,sT]lt12(κ2+1+λ)gds.This condition implies that ddwFλ(γw)=01<ssW,sT><sW,2(κ2+1)T>+ +<R(sT,T)T,W>+12<sW,(κ2+1+λ)T>ds=0LE,W>ds+[<sW,sT>+<W,(s)2T+ΛT>]0L,where E=(s)3Ts(ΛT)+R(sT,T)T,g=<TW,T>and Λ=λ3κ242.Thus, we can state the following.

Lemma 2.10: Let γ be one parameterfamily curves in de Sitter space S13.ifγω is inextensible flow of space-like γ in de Sitter space S13, then g=<sW,T>=π2κδ(γ)+π1s,where π1, π2 are smooth functions of time and arc-length.

Theorem 2.11: Let γ be one parameterfamily curves in de Sitter space S13.ifγw is inextensible flow of space-like γ in de Sitter space S13, then W(κ2)=2π1s+κ[s[π1κ+π3τ+π2s]+τ[π2T+π3s]]+4g(κ2+1)+π2κ,where π1, π2, π3 are smooth functions of time and arc-length.

proof: Firstly, we obtain swT=π1sγ[π1[π1κ+π3τ+π2s]κδ(γ)]T+[s[π1κ+π3τ+π2s]+τ[π2T+π3s]]N+[s[π2τ+π3s]+τ[π1κ+π3τ+π2s]]B.Since, we immediately arrive at R(γs,γt)T=π2N+π3B.Therefore, W(κ2)=2<TTW,TT>+4g(κ2+1)+2<R(W,T)T,TT>=2π1s+κ[s[π1κ+π3τ+π2s]+τ[π2τ+π3s]]+4g(κ2+1)+π2κ.Now, we can obtain following equation in terms of flows.

Lemma 2.12: E=(1κ2δ(γ)+Λ)γ+[3κsκδ(γ)Λs]T+(2Ks2κ+κ3δ(γ)+κτ2κΛκ)N+(2κsτ+κτs)B.

Theorem 2.13: Let γ be one parameterfamily curves in de Sitter space S13.ifγω is inextensible flow of space-like γ in de Sitter space S13, then wE=[w(1κ2δ(γ)+Λ)π1[3κsκδ(γ)Λs]1κπ1s(2Ks2κ+κ3δ(γ)+κτ2κΛκ)+(2κsτ+κτs)1τ[π1κδ(γ)+s[1κπ1s][1κ[[π1κ+π3τ+π2s]κδ(γ)]]]]γ+[π1(1κ2δ(γ)+Λ)+w[3κsκδ(γ)Λs]+1κ[[π1κ+π3T+π2s]κδ(γ)](2Ks2κ+κ3δ(γ)+κτ2κΛκ)+1τ[π2+s[1κ[[π1κ+π3τ+π2s]κδ(γ)]]1κπ1stκδ(γ)](2κsτ+κτs)]T+[π2(1κ2δ(γ)+Λ)+[π1κ+π3τ+π2s][3κsκδ(γ)Λs+w(2Ks2κ+κ3δ(γ)+κτ2κΛκ)+(2κsτ+κτs)1τ[κ[1κ[[π1κ+π3τ+π2s]κδ(γ)]]+τ[1κ[s[π2T+π3s]+τ[π1κ+π3τ+π2s]+2π3]]κδ(γ)[π1κ+π3τ+π2s]]]N+[π3(1κ2δ(γ)+Λ)+[3κsκδ(γ)Λs][π2T+π3s]+1κ(2Ks2κ+κ3δ(γ)+κτ2κΛκ)[s[π2T+π3s]+τ[π1κ+π3τ+π2s]+2π3]+w(2κsτ+κτs)]B.

Example 2.14: The time-helix is parametrized by γ(s,w)=(A(w)cos(s),A(w)sin(s),B(w)s,0), where $A, B$ are functions only of time.Projection of γ at xyz-plane:

Fig. 1

Time-helix is illustrated using colours Magenta, Cyan, Green at the time t=1.2, t=1.8, t=2.2, respectively

Fig. 2

Time-helix is illustrated using colours Magenta, Cyan, Green at the time t=1.2, t=1.8, t=2.2, respectively


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About the article

Received: 2016-06-22

Accepted: 2016-08-24

Published Online: 2016-11-25

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0071. Export Citation

© 2016 Yeneroğlu, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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