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

Mustafa Yeneroğlu

doi:10.1515/math-2016-0071

Open Access Published by De Gruyter Open Access November 25, 2016

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

Mustafa Yeneroğlu

From the journal Open Mathematics

https://doi.org/10.1515/math-2016-0071

Abstract

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 dS_n, 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, [1–6].

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 [16–20].

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, [8–14].

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={x∈R14|x,x=1}.

It is well-known that to each unit speed space-like curve γ:I→S13 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.

∂γ∂w=π1T+π2N+π3B

Putting

W=W(w,t)=∂γ∂w,V(u,w)=∂γ∂u=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 ∂γ∂w in de Sitter space S13 is said to be inextensible if

∂∂w∥∂α∂u∥=0.(2)

Theorem 2.2

Let ∂γ∂w. be a smooth flow of γ. The flow is inextensible if and only if

π2vκδ(γ)+∂π1∂u=0(3)

Now, assume that γ is arc-length parametrized curve. Then, we have

Lemma 2.3

∇wT=−π1γ+[π1κ+π3τ+∂π2∂s]N+[π2τ+∂π3∂s]B,

where π₁, π₂, π₃are smooth functions of time and arc-length.

Proof

From the definition of inextensible flow, we have

∇wT=−π1γ+[π2κδ(γ)+∂π1∂s]T+[π1κ+π3τ+∂π2∂s]N+[π2τ+∂π3∂s]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 spaceS13.if∂γ∂ωis inextensible flow of space-like γ in de Sitter spaceS13, then the evolution of κ is given by

∂κ∂w=∂∂s[π1κ+π3τ+∂π2∂s]+τ[π2T+∂π3∂s]+1κ∂π1∂s+π2,

where π₁, π₂, π₃are smooth functions of time and arc-length.

Proof

A differentiation in Eq. (4) and the Frenet formulas give us that

∇s∇wT=−∂π1∂sγ−[π1−[π1κ+π3τ+∂π2∂s]κδ(γ)]T+[∂∂s[π1κ+π3τ+∂π2∂s]+τ[π2τ+∂π3∂s]]N+[∂∂s[π2T+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]]B.

Using the formula of the curvature, we write a relation

∇w∇sT−∇s∇wT=R(∂γ∂s,∂γ∂w)T.

We immediately arrive at

R(∂γ∂s,∂γ∂w)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)X2−g(X2,X3)X1.

Then,

R(T,N)T=g(T,T)N−g(N,T)T=N,R(T,B)T=g(T,T)B−g(B,T)T=B.

From above equations, we get

R(∂γ∂s,∂γ∂w)T=π2N+π3B.

Then, we can write

∇w∇sT=∇s∇wT+R(∂γ∂s,∂γ∂w)T.

Thus it is easy to obtain that

∇w∇sT=−∂π1∂sγ−[π1−[π1κ+π3τ+∂π2∂s]κδ(γ)]T

+[∂∂s[π1κ+π3τ+∂π2∂s]+τ[π2τ+∂π3∂s]+π2]N+[∂∂s[π2τ+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]+π3]B.

On the other hand, we have

∂κ∂w=∂∂wg(∇sT,N).

Since, we express

∂κ∂w=g(∇w∇sT,N)+g(∇sT,∇wN).

Moreover, by the definition of metric tensor, we have

g(N,∇wN)=0.

Then

g(∇sT,∇wN)=−g(γ,∇wN)=1κ∂π1∂s.

Combining these we have

∂κ∂w=∂∂s[π1κ+π3τ+∂π2∂s]+τ[π2κ+∂π3∂s]+1κ∂π1∂s+π2.

Thus, we obtain the theorem. This completes the proof.

From the above theorem, we have

Theorem 2.5

∇wN=−1κ∂π1∂sγ+1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]T+1κ[∂∂s[π2τ+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]+2π3]B,

where π₁, π₂, π₃are smooth functions of time and arc-length.

Proof

Using Frenet equations, we have

∇w∇sT=−∂γ∂w+∂κ∂wN+κ∇wN.

Then,

κ∇wN=−∂π1∂sγ+[[π1κ+π3τ+∂π2∂s]κδ(γ)]T+[−∂κ∂t+∂∂s[π1κ+π3τ+∂π2∂s]+τ[π2τ+∂π3∂s]+2π2]N+[∂∂s[π2τ+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]+2π3]B.

Therefore

g(N,∇wN)=0.

From above equation we obtain

∇wN=−1κ∂π1∂sγ+1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]T+1κ[−∂κ∂t+∂∂s[π1κ+π3τ+∂π2∂s]+τ[π2τ+∂π3∂s]+2π2]N+1κ[∂∂s[π2τ+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]+2π3]B,

which completes the proof.

Theorem 2.6

Let γ be one parameterfamily curves in de Sitter spaceS13.If∂γ∂wis inextensible flow of space-like γ in de Sitter spaceS13then

∇wB=1τ[π1κδ(γ)+∂∂s[−1κ∂π1∂s]−[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]]γ+1τ[−π2+∂∂s[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]−1κ∂π1∂s−∂∂tκδ(γ)]T+1τ[κ[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]+τ[1κ[∂∂s[π2T+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]+2π3]]−κδ(γ)[π1κ+π3τ+∂π2∂s]]N,

where π₁, π₂, π₃are smooth functions of time and arc-length.

Proof

Assume that ∂γ∂w be inextensible flow of γ.

∇s∇wN=[∂∂s[−1κ∂π1∂s]−[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]]γ+[∂∂s[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]−1κ∂π1∂s]T+[κ[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]+τ[1κ[∂∂s[π2τ+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]+2π3]]]N+∂∂s[1κ[∂∂s[π2τ+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]+2π3]]B.

Under the assumption of space-like curve, we have

∇w∇sN=−π1κδ(γ)γ+∂∂tκδ(γ)T+κδ(γ)[π1κ+π3τ+∂π2∂s]N+[κδ(γ)[π2τ+∂π3∂s]+∂τ∂t]B+τ∇wB.

Using the formula of the curvature, we write a relation

∇w∇sN−∇s∇wN=R(∂γ∂s,∂γ∂w)N.

Thus, it is seen that

R(∂γ∂s,∂γ∂w)N=π2R(T,N)N+π3R(T,B)N.

By using formula of curvature, we have

R(T,N)N=g(T,N)N−g(N,N)T=−T,R(T,B)N=g(T,N)B−g(B,N)T=0.

Arranging the last equations, we obtain

R(∂γ∂s,∂γ∂w)N=−π2T.

Therefore, we can easily see that

τ∇wB=[π1κδ(γ)+∂∂s[−1κ∂π1∂s]−[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]]γ+[−π2+∂∂s[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]−1κ∂π1∂s−∂∂tκδ(γ)]T+[κ[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]+τ[1κ[∂∂s[π2τ+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]+2π3]]−κδ(γ)[π1κ+π3τ+∂π2∂s]]N

+[∂∂s[1κ[∂∂s[π2T+∂π3∂s]+τ[π1K+π3τ+∂π2∂s]+2π3]]−[κδ(γ)[π2T+∂π3∂s]+∂τ∂t]]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 spaceS13.if∂γ∂tis inextensible flow of space-like γ in de Sitter spaceS13then the evolution of τ is given by

∂τ∂w=∂∂s[1κ[∂∂s[π2T+∂π3∂s]+τ[π1K+π3τ+∂π2∂s]+2π3]]−κδ(γ)[π2T+∂π3∂s],

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=∥∇TT∥2.

Theorem 2.8 (Main Theorem)

W(κ2+1)=2<∇s∇sW,∇sT>+4g(κ2+1)+2<R(W,T)T,∇sT>,

whereg=−<∇TW,T>.

proof

From Euler equations, we easily have

W(κ2+1)=2<∇s∇sW,∇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)=12∫0Lκ2+1+λds=12∫01(∥∇sT∥2+λ)v(t)dt.

For a variation γ_ω with variation field M, we compute

ddwFλ(γw)=12∫01W(κ2+1)v+(κ2+1+λ)W(v)dt=12∫01W(κ2+1)−(κ2+1+λ)gds=∫01<∇s∇sW,∇sT<+2g(κ2+1)+>R(W,T)T,∇sT]lt−12(κ2+1+λ)gds.

This condition implies that

ddwFλ(γw)=∫01<∇s∇sW,∇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)3T−∇s(ΛT)+R(∇sT,T)T,g=−<∇TW,T>

and

Λ=λ−3κ2−42.

Thus, we can state the following.

Lemma 2.10

Let γ be one parameterfamily curves in de Sitter spaceS13.if∂γ∂ωis inextensible flow of space-like γ in de Sitter spaceS13, then

g=−<∇sW,T>=π2κδ(γ)+∂π1∂s,

where π₁, π₂are smooth functions of time and arc-length.

Theorem 2.11

Let γ be one parameterfamily curves in de Sitter spaceS13.if∂γ∂wis inextensible flow of space-like γ in de Sitter spaceS13, then

W(κ2)=2∂π1∂s+κ[∂∂s[π1κ+π3τ+∂π2∂s]+τ[π2T+∂π3∂s]]+4g(κ2+1)+π2κ,

where π₁, π₂, π₃are smooth functions of time and arc-length.

proof

Firstly, we obtain

∇s∇wT=−∂π1∂sγ−[π1−[π1κ+π3τ+∂π2∂s]κδ(γ)]T+[∂∂s[π1κ+π3τ+∂π2∂s]+τ[π2T+∂π3∂s]]N+[∂∂s[π2τ+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]]B.

Since, we immediately arrive at

R(∂γ∂s,∂γ∂t)T=π2N+π3B.

Therefore,

W(κ2)=2<∇T∇TW,∇TT>+4g(κ2+1)+2<R(W,T)T,∇TT>=2∂π1∂s+κ[∂∂s[π1κ+π3τ+∂π2∂s]+τ[π2τ+∂π3∂s]]+4g(κ2+1)+π2κ.

Now, we can obtain following equation in terms of flows.

Lemma 2.12

E=(1−κ2δ(γ)+Λ)γ+[3∂κ∂sκδ(γ)−∂Λ∂s]T+(∂2K∂s2−κ+κ3δ(γ)+κτ2−κΛ−κ)N+(2∂κ∂sτ+κ∂τ∂s)B.

Theorem 2.13

Let γ be one parameterfamily curves in de Sitter spaceS13.if∂γ∂ωis inextensible flow of space-like γ in de Sitter spaceS13, then

∇wE=[∂∂w(1−κ2δ(γ)+Λ)−π1[3∂κ∂sκδ(γ)−∂Λ∂s]−1κ∂π1∂s(∂2K∂s2−κ+κ3δ(γ)+κτ2−κΛ−κ)+(2∂κ∂sτ+κ∂τ∂s)1τ[π1κδ(γ)+∂∂s[−1κ∂π1∂s]−[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]]]γ+[π1(1−κ2δ(γ)+Λ)+∂∂w[3∂κ∂sκδ(γ)−∂Λ∂s]+1κ[[π1κ+π3T+∂π2∂s]κδ(γ)](∂2K∂s2−κ+κ3δ(γ)+κτ2−κΛ−κ)+1τ[−π2+∂∂s[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]−1κ∂π1∂s−∂∂tκδ(γ)](2∂κ∂sτ+κ∂τ∂s)]T+[π2(1−κ2δ(γ)+Λ)+[π1κ+π3τ+∂π2∂s][3∂κ∂sκδ(γ)−∂Λ∂s+∂∂w(∂2K∂s2−κ+κ3δ(γ)+κτ2−κΛ−κ)+(2∂κ∂sτ+κ∂τ∂s)1τ[κ[1κ[[π1κ+π3τ+∂π2∂s]κδ(γ)]]+τ[1κ[∂∂s[π2T+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]+2π3]]−κδ(γ)[π1κ+π3τ+∂π2∂s]]]N+[π3(1−κ2δ(γ)+Λ)+[3∂κ∂sκδ(γ)−∂Λ∂s][π2T+∂π3∂s]+1κ(∂2K∂s2−κ+κ3δ(γ)+κτ2−κΛ−κ)[∂∂s[π2T+∂π3∂s]+τ[π1κ+π3τ+∂π2∂s]+2π3]+∂∂w(2∂κ∂sτ+κ∂τ∂s)]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 atxyz-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. 1

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

$Fig. 2 Time-helix is illustrated using colours Magenta, Cyan, Green at the time t=1.2, t=1.8, t=2.2, respectively$

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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Received: 2016-6-22

Accepted: 2016-8-24

Published Online: 2016-11-25

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Abstract

1 Introduction

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

References

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