Inextensible Flows of Spacelike Curves on Spacelike Surfaces

(3s.) v. 31 2 (2013): 9–17. ISSN-00378712 in press doi:10.5269/bspm.v31i2.15754

Bol. Soc. Paran. Mat. c

SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm

Inextensible Flows of Spacelike Curves on Spacelike Surfaces according to Darboux Frame in M31 Selçuk Baş and Talat Körpınar

abstract: In this paper, we study inextensible flows of spacelike curves on oriented spacelike surfaces in M31 . We give necessary and sufficient conditions for inextensible flows of spacelike curves on oriented spacelike surfaces in M31 .

Key Words: Inextensible flows, Darboux Frame, Curvatures. Contents 1 Introduction

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2 Preliminaries

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3 Inextensible Flows of Spacelike Curves on Spacelike Surface according to Darboux Frame in M31

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1. Introduction Construction of fluid flows constitutes an active research field with a high industrial impact. Corresponding real-world measurements in concrete scenarios complement numerical results from direct simulations of the Navier-Stokes equation, particularly in the case of turbulent flows, and for the understanding of the complex spatio-temporal evolution of instationary flow phenomena. More and more advanced imaging devices (lasers, highspeed cameras, control logic, etc.) are currently developed that allow to record fully timeresolved image sequences of fluid flows at high resolutions. As a consequence, there is a need for advanced algorithms for the analysis of such data, to provide the basis for a subsequent pattern analysis, and with abundant applications across various areas. This study is organised as follows: Firstly, we study inextensible flows of spacelike curves on oriented spacelike surfaces in M31 . Finally, we give necessary and sufficient conditions for inextensible flows of spacelike curves on oriented spacelike surfaces in M31 . 2. Preliminaries The Minkowski 3-space M31 provided with the standard flat metric given by h, i = −dx21 + dx22 + dx23 , 2000 Mathematics Subject Classification: 53A35

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Typeset by BSM P style. c Soc. Paran. de Mat.

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Selçuk Baş and Talat Körpınar

3 where (x1 , x2 , x3 ) is a rectangular coordinate system p of M1 . Recall that, the norm 3 of an arbitrary vector a ∈ M1 is given by kak = ha, ai. γ is called a unit speed curve if velocity vector v of γ satisfies kak = 1. Denote by {T, N, B} the moving Frenet–Serret frame along the spacelike curve γ in the space M31 . For an arbitrary spacelike curve γ with first and second curvature, κ and τ in the space M31 , the following Frenet–Serret formulae is given

T′

= κN

N′ B′

= κT + τ B = τ N,

(2.1)

where hT, Ti =

1, hN, Ni = −1, hB, Bi = 1,

hT, Ni = hT, Bi = hN, Bi = 0. Here, curvature functions are defined by κ = κ(s) and τ = τ (s). Torsion of the spacelike curve γ is given by the aid of the mixed product τ=

[γ ′ , γ ′′ , γ ′′′ ] . κ2

A surface M in the Minkowski 3-space M31 is said to be space-like, time-like surface if, respectively the induced metric on the surface is a positive definite Riemannian metric, Lorentz metric. In other words, the normal vector on the spacelike (timelike) surface is a timelike (spacelike) vector [9]. If the surface M is an oriented spacelike surface, then the curve α(s) lying on M is a spacelike curve. Thus, the equations which describe the Darboux frame of α(s) is given by : T′ ′

P n′

= κg P + κn n, = −κg T + τ g n, = κn T+τ g P,

(2.2)

where T, P,n satisfy the following properties: < T, T > = 1, < n, n > = −1, < P, P > = 1, < T, n > =< T, P > =< n, P > = 0. In this frame T is the unit tangent of the curve, n is the unit normal of the surface M and P is a unit vector given by P = n × T.

Inextensible Flows of Spacelike Curves

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3. Inextensible Flows of Spacelike Curves on Spacelike Surface according to Darboux Frame in M31 Let α (u, t) is a one parameter family of smooth spacelike curves in M31 . The arclength of α is given by Zu ∂α s(u) = du, ∂u

(3.1)

0

where

The operator

1 ∂α ∂α ∂α 2 = ∂u ∂u , ∂u .

(3.2)

∂ is given in terms of u by ∂s

1 ∂ ∂ = , ∂s ν ∂u ∂α where v = and the arclength parameter is ds = vdu. ∂u Any flow of α can be represented as ∂α D D = AD 1 T + A2 P + A3 n, ∂t

(3.3)

D D where AD 1 , A2 , A3 are smooth functions. Letting the arclength variation be

s(u, t) =

Zu

vdu.

0

M31

In the the requirement that the curve not be subject to any elongation or compression can be expressed by the condition ∂ s(u, t) = ∂t

Zu

∂v du = 0, ∂t

(3.4)

0

for all u ∈ [0, l] . Definition 3.1. Let M be an oriented spacelike surface and α lying on M in ∂α on M are said to be inextensible if Minkowski 3-space M31 . The flow ∂t ∂ ∂α = 0. ∂t ∂u

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Lemma 3.2. Let M be an oriented spacelike surface and α lying on M in Minkowski ∂α D D = AD 3-space M31 . The flow 1 T + A2 P + A3 n is inextensible if and only if ∂t ∂v ∂AD 1 D − = −AD 2 vκg + A3 vκn . ∂t ∂u

(3.5)

∂α be a smooth flow of the spacelike curve α. Using definition Proof: Suppose that ∂t of α, we have ∂α ∂α 2 . (3.6) , v = ∂u ∂u ∂ ∂ and commute since and are independent coordinates. ∂u ∂t Further differentiation of (3.6) gives ∂v ∂ ∂α ∂α 2v . = , ∂t ∂t ∂u ∂u ∂ ∂ and , we have ∂u ∂t ∂v ∂α ∂ ∂α v = , ( ) . ∂t ∂u ∂u ∂t

On the other hand, changing

From (3.3), we obtain v

∂v = ∂t

∂α ∂ D D , AD T + A P + A . n 1 2 3 ∂u ∂u

By the formula of the Darboux, we have D ∂A1 ∂AD ∂v 2 D D D D =< T, − A2 vκg + A3 vκn T + A1 vκg + + A3 vτ g P ∂t ∂u ∂u ∂AD 3 D + AD n >. 1 vκn + A2 vτ g + ∂u Making necessary calculations from above equation, we have (3.5), which proves the lemma. 2 Theorem 3.3. Let M be an oriented spacelike surface and α lying on M in ∂α is inextensible if and only if Minkowski 3-space M31 . The flow ∂t ∂AD 1 D = AD 2 vκg − A3 vκn . ∂u

(3.7)


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∂α be inextensible. ∂t ∂α is inextensible if and only if By Definition 3.1, the flow ∂t

Proof: Assume that

∂ s(u, t) = ∂t

Zu

∂v du = ∂t

0

Zu

∂AD 1 D D − A2 vκg + A3 vκn du = 0. ∂u

(3.8)

0

Substituting (3.5) in (3.8) complete the proof of the theorem.

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We now restrict ourselves to arc length parametrized curves. That is, v = 1 and the local coordinate u corresponds to the curve arc length s. We require the following lemma. Lemma 3.4. ∂AD ∂AD 2 3 D D D AD κ + + A τ n, (3.9) P + A κ + A τ + 1 g 3 g 1 n 2 g ∂s ∂s ∂P ∂AD 2 D D = − A1 κg + + A3 τ g T + ψn, (3.10) ∂t ∂s ∂n ∂AD 3 D = AD T − ψP, (3.11) 1 κn + A2 τ g + ∂t ∂s ∂P ,n . where ψ = ∂t ∂T ∂t

=

Proof: Using definition of α, we have ∂T ∂ ∂α ∂ D = = (AD T + AD 2 P + A3 n). ∂t ∂t ∂s ∂s 1 Using the Darboux equations, we have ∂T ∂t

∂AD ∂AD 1 2 D D D D = − A2 κg + A3 κn T + A1 κg + + A3 τ g P ∂s ∂s ∂AD 3 D n. (3.12) + AD κ + A τ + n g 1 2 ∂s

Thus, we rewrite (3.12) as follows: ∂T = ∂t

AD 1 κg

∂AD ∂AD 2 3 D D D + A3 τ g P + A1 κn + A2 τ g + n. + ∂s ∂s

Considering (3.12), we have

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∂P ∂AD 2 D = 0, + A3 τ g + T, + ∂s ∂t ∂n ∂AD 3 D −AD + T, = 0, κ − A τ − n g 1 2 ∂s ∂t ∂n = 0. ψ + P, ∂t AD 1 κg

Then, a straightforward computation using above system gives ∂P ∂t ∂n ∂t

∂AD 2 D = − AD + A κ + τ 1 g 3 g T + ψn, ∂s ∂AD 3 D D T − ψP, = A1 κn + A2 τ g + ∂s

∂P where ψ = ,n . ∂t Thus, we obtain the theorem.

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The following theorem states the conditions on the curvature and torsion for the flow to be inextensible. Theorem 3.5. Let M be an oriented spacelike surface and α lying on M in ∂α Minkowski 3-space M31 . If is inextensible, then the following system of partial ∂t differential equations holds: ∂κg − κn ψ ∂t ∂κn + κg ψ ∂t

= =

∂ ∂ 2 AD ∂ ∂AD 2 3 D D D 2 (AD + (A τ ) + A κ τ + A τ + τ g, g n g 1 κg ) + 3 1 2 g ∂s ∂s2 ∂s ∂s ∂ ∂ ∂ 2 AD ∂AD 3 2 2 (AD (AD + AD τ g + AD 1 κn ) + 2 τ g) + 1 κg τ g + 3 τ g. 2 ∂s ∂s ∂s ∂s

Proof: Using (3.9), we have ∂ ∂T ∂s ∂t

= =

∂A D ∂ ∂A D D D D 3 2 P + A κ + A τ + AD κ + + A τ n n g g g 1 2 1 3 ∂s ∂s ∂s ∂A D ∂A D D 2 3 + AD κg ]T κn − A D [ AD 3 τg 1 κg + 1 κn + A 2 τ g + ∂s ∂s ∂ ∂ 2 AD ∂ ∂A D D D D 2 2 3 + (A D κ ) + + (A τ ) + A κ τ + A τ + τ P g 1 g 3 g 1 n g 2 g ∂s ∂s2 ∂s ∂s ∂ ∂ 2 AD ∂A D ∂ D D 2 3 2 κ τ + τ n. (A D (A D + A τ + A + g g g g 1 κn ) + 2 τ g) + 1 3 ∂s ∂s ∂s2 ∂s


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On the other hand, from Darboux frame we have ∂ ∂T ∂t ∂s

∂ (κg P + κn n) ∂t ∂AD ∂AD 3 2 D D D D + A3 τ g − κn A1 κn + A2 τ g + ]T = −[κg A1 κg + ∂s ∂s ∂κn ∂κg + κg ψ + − κn ψ P. n+ ∂t ∂t =

Thus, ∂κn ∂ ∂ ∂ 2 AD ∂AD 3 2 D D 2 + κg ψ = (AD (A + A τ g + AD κ ) + τ ) + κ τ + n g g g 1 3 τg ∂t ∂s 1 ∂s 2 ∂s2 ∂s and ∂ ∂ 2 AD ∂ ∂AD ∂κg 2 3 D D D 2 − κn ψ = (AD κ ) + + (A τ ) + A κ τ + A τ + τ g. g g n g 1 2 g ∂t ∂s 1 ∂s2 ∂s 3 ∂s Thus, we obtain the theorem. 2 Corollary 3.6. ∂A D ∂κn ∂ D ∂ 2 AD ∂ 2 2 3 − AD τ g + AD =− (A D A2 τ g + +κg ψ. 1 κg τ g + 3 τg + 1 κn ) + ∂s ∂t ∂s ∂s ∂s2

Proof: Similarly, we have ∂ ∂AD ∂ ∂n 3 D = − AD T − ψP κ + A τ + 1 n 2 g ∂s ∂t ∂s ∂s 2 ∂ AD ∂ ∂ 3 D D A2 τ g + + κg ψ]T (A κn ) + = [− ∂s 1 ∂s ∂s2 ∂ψ ∂AD 3 D +[−κg AD − ]P 1 κn + A2 τ g + ∂s ∂s ∂AD 3 D −[κn AD κ + A τ + + τ g ψ ]n. n g 1 2 ∂s On the other hand, a straight forward computation gives ∂ ∂n ∂AD ∂κn 2 D 2 = [− AD τ + A ]T κ τ + τ g g g 1 3 g + ∂t ∂s ∂s ∂t ∂τ g ∂AD 2 D κ + A ]P κ τ +[ AD κ κ + n 3 n g + 1 g n ∂s ∂t ∂AD 3 D κ + A τ + +κn AD + ψτ g n. 1 n 2 g ∂s Combining these we obtain the corollary.

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In the light of Theorem 3.5, we express the following lemmas without proofs:

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Lemma 3.7. AD 1 κg κn +

∂ψ ∂τ g ∂AD ∂AD 3 2 D D − κn + AD = −κ . κ τ + A κ + A τ + g 3 n g 1 n 2 g ∂s ∂t ∂s ∂s

Corollary 3.8. Let M be an oriented spacelike surface, α lying on M and the flow ∂α is inextensible in Minkowski 3-space M31 . If α is a geodesic curve, then ∂t ∂AD ∂τ g ∂ψ = − 2 κn − AD . 3 κn τ g − ∂s ∂s ∂t Proof: By using κg = 0 in Lemma 3.7, we get above equation. This completes the proof. 2 Corollary 3.9. Let M be an oriented spacelike surface, α lying on M and the flow ∂α is inextensible in Minkowski 3-space M31 . If α is a principal line, then ∂t ∂ψ ∂AD ∂AD ∂τ g 3 2 D κg + + κn + = −AD 1 κg κn − κg A1 κn , ∂s ∂s ∂s ∂t Proof: Substituting τ g = 0 in Lemma 3.7, we get above equation. This completes the proof. 2 Corollary 3.10. Let M be an oriented spacelike surface, α lying on M and the ∂α is inextensible in Minkowski 3-space M31 . If α is a asymptotic line, then flow ∂t ∂ψ ∂AD 3 D + = 0. κg A2 τ g + ∂s ∂s Proof: By using κn = 0 in Lemma 3.7, we get above equation. This completes the proof. 2 Acknowledgments The authors wish to thank the referee for providing constructive comments and valuable suggestions. References 1. U. Abresch, J. Langer: The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), 175-196. 2. B. Andrews: Evolving convex curves,Calculus of Variations and Partial Differential Equations, 7 (1998), 315-371. 3. M. Babaarslan, Y. Yayli: The characterizations of constant slope surfaces and Bertrand curves, International Journal of the Physical Sciences 6(8) (2011), 1868-1875.


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Selçuk Baş Fırat University, Department of Mathematics, 23119 Elazığ, Turkey E-mail address: [email protected] and Talat Körpınar Fırat University, Department of Mathematics, 23119 Elazığ, Turkey E-mail address: [email protected]