Inextensible Flows of Curves in Lie Groups

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1 CJMS. 113, 3-3 Caspian Journal of Mathematical Sciences CJMS University of Mazandaran, Iran ISSN: Inextensible Flows of Curves in Lie Groups Gökmen Yıldız a and O. Zeki Okuyucu b, 1 a Department of Mathematics, University of Bilecik Şeyh Edebali, Bilecik, Turkey b Department of Mathematics, University of Bilecik Şeyh Edebali, Bilecik, Turkey Abstract In this paper, we study inextensible flows of curves in three dimensional Lie groups with a bi-invariant metric. The necessary and sufficient conditions for inextensible curve flow are expressed as a partial differential equation involving the curvatures. Also, we give some results for special cases of Lie groups. Keywords: Inextensible flows; Lie groups 1. INTRODUCTION Flows of curves are used in engineering applications such as modeling ship hulls, buildings, airplane wings, garments, ducts, automobile parts. Moreover Chirikjian and Burdick explain the kinematics of hyperredundant or serpentine robot as the flow of plane curve in [1]. Inextensible flows of curves and developable surfaces in Euclidean 3- space have been studied by Kwon and Park in [8]. The flow of a curve is said to be inextensible if, its arc-length is preserved. Inextensible flows of curves have great importance in computer vision and computer animation in addition to the structural mechanics see [4],[7], [9]. Inextensible flows of curves have been studied in many studies. For example, Yıldız et al. [1] have investigated inextensible flows of curves 1 Corresponding author: osman.okuyucu@bilecik.edu.tr, Received: 8 Jan 13 Revised: 9 Mar 13 Accepted: 11 July 13

2 4 Gökmen Yıldız, O. Zeki Okuyucu according to darboux frame in Euclidean 3-space. Recently, Gürbüz [6] have introduced inextensible flows of spacelike, timelike and null curves in Minkowski space. Quite recently, Öğrenmiş et al. [11] have examined inextensible curves in Galilean space and etc. In the literature, there are many papers about curves in Lie groups. For example, Çiftçi [3] has defined general helices in three dimensional Lie groups with a bi-invariant metric and obtained a generalization of Lancret s theorem and given a relation between the geodesics of the so-called cylinders and general helices. Also, Okuyucu et al. [1] have studied slant helices in Lie groups. Further, AWk-type general helices in Lie groups have been given by Yoon in [13]. In the present paper, we study inextensible flows of curves in three dimensional Lie groups with a bi-invariant metric and we obtain some results for special cases of three dimensional Lie groups. So we hope that this study contributes to the curves theory in Lie groups.. Preliminaries Let G be a Lie group with a bi-invariant metric, and D be the Levi-Civita connection of Lie group G. If g denotes the Lie algebra of G then we know that g is isomorphic to T e G, where e is neutral element of G. If, is a bi-invariant metric on G then we will have that and X, [Y, Z] = [X, Y ], Z D X Y = 1 [X, Y ] for all X, Y and Z g. Let α : I R G be an arc-lenghted curve and {X 1, X,..., X n } be an orthonormal basis of g. In this case, we write that any two vector fields W and Z along the curve α as W = n i=1 w ix i and Z = n i=1 z ix i, where w i : I R and z i : I R are smooth functions for 1 i n. Also the Lie bracket of two vector fields W and Z is given by n [W, Z] = w i z i [X i, X j ] i=1 and the covariant derivative of W along the curve α with the notation D α W is given by D α W = W 1 [T, W ],.1 where T = α and W = n i=1 w i X i or W = n i=1 dw dt X i. Note that if W is the restriction of a left-invariant vector field to the curve α, then W = see [] for details.

3 Inextensible Flows of Curves in Lie Groups 5 Let G be a three dimensional Lie group and T, N, B, κ, τ denote the Frenet apparatus of the curve α, then we calculate κ = T. Definition.1. [3] Let α : I R G be an arc-length parameterized curve with the Frenet apparatus T, N, B, κ, τ. Then τ G = 1 [T, N], B or τ G = 1 κ T, [T, T ] 1 τ 4κ [T, T ]. τ Proposition.. [1] Let α : I R G be an arc-length parameterized curve with the Frenet apparatus {T, N, B}. Then the following equalities hold. [T, N] = [T, N], B B = τ G B [T, B] = [T, B], N N = τ G N Remark.3. [3] and [5] Let G be a Lie group with a bi-invariant metric,. Then the following equalities can be given in different Lie groups. i If G is Abelian group then τ G =. ii If G is SO 3 then τ G = 1. iii If G is SU then τ G = 1. Let α : I R G be a curve with arc-length parameter s. For Frenet vectors of the curve α, we have by.1 and Proposition. that dt ds dn ds db ds = κ κ τ τ G T N,. τ τ G B where {T, N, B} is Frenet frame, τ G = 1 [T, N], B and κ, τ are curvatures of the curve α in three dimensional Lie group G, respectively. 3. Inextensible Flows of Curves in Lie Groups Throughout this paper, unless otherwise stated, we assume that α : [, l] [, w G is a one - parameter family of smooth curves in three dimensional Lie group G, where l is the arc-length of the initial curve. Let u be the curve parametrization variable, u l. If the speed of the curve α is given by v = dα, then the arc-length of α is given as a function of u by du su = u α u du = u vdu.

4 6 Gökmen Yıldız, O. Zeki Okuyucu The operator is given by = 1 v u. 3.1 In this case; the arc-length parameter is ds = vdu. Let αu, t : [, l] [, w G be a differentiable curve with the Frenet vector field {T, N, B} in three dimensional Lie group G. Any flow of the curve can be expressed as follows α = ft gn hb, where f, g and h are scalar speed functions of the curve α. Let the arc-length variation be su, t = u vdu. In three dimensional Lie group G, the requirement that the curve not be subject to any elongation or compression can be expressed by the condition where u [, l]. su, t = u v du =, 3. in three di- Definition 3.1. A curve evolution αu, t and its flow α mensional Lie group G are said to be inextensible if α u =. Before deriving the necessary and sufficient condition for inelastic curve flow, we need the following lemma. Lemma 3.. Let αu, t be a differentiable curve with the Frenet vector field {T, N, B} and the smooth flow of the curve αu, t be α = ft gn hb in three dimensional Lie group G. Then we have the following equality: v = f gvκ. 3.3 u

5 Inextensible Flows of Curves in Lie Groups 7 Proof. Since u and are commutative and v = α u, α u, we have showing. that v v = α u, α u α = u, α u α = u, ft gn hb u = < vt, f g T fvκn u u N g vκt vτ τ GB h u B f h vτ τ G N >= v u gvκ. This leads us to the consequence of the equality below. v = f u gvκ. Theorem 3.3. Let α = ft gn hb be a smooth flow of the curve α in three dimensional Lie group G. The flow is inextensible if and only if f = gκ. 3.4 Proof. Let us assume that the curve flow is inextensible. 3. and 3.3, we have su, t = u v du = which leads us to the fact that f gvκ =. u By using the last equation with 3.1, we get u f u gvκ du =, Combining f = gκ. On the contrary, following similar way as above, the proof can be completed. Now, suppose that the curve α is an arc-length parametrized curve in three dimensional Lie group G. That is, v = 1 and the local coordinate u corresponds to the curve arclength s.

6 8 Gökmen Yıldız, O. Zeki Okuyucu Lemma 3.4. Let α be a arc-length parameterized curve with the Frenet vectors {T, N, B} in three dimensional Lie group G. If the flow of the curve α is inextensible, then the derivatives of {T, N, B} with respect to t are = fκ g hτ τ G N = fκ g hτ τ G T ψb, B = where ψ =, B. gτ τ G h T ψn, gτ τ G h B, Proof. For and are commutative, we have = α = α = ft gn hb = f g T fκn N g κt τ τ GB h B h τ τ GN f = gκ T fκ g hτ τ G N gτ τ G h B. Substituting 3.4 into the last equation, we get = fκ g hτ τ G N gτ τ G h Also, since T, N = T, B = N, B =, we get = T, N =, N T, = fκ g = T, B =, B T, B = = N, B =, B N, B = ψ B. hτ τ G gτ τ G h N, B From the above equations, we obtain that = fκ g hτ τ G T ψb, and B = gτ τ G h T ψn.. T, B T,,,

7 Inextensible Flows of Curves in Lie Groups 9 Lemma 3.4 includes the following special cases for derivatives of {T, N, B} with respect to t: i If G is Abelian group then τ G =. Thus we have that = fκ g hτ N = fκ g hτ B = gτ h T ψn. gτ h T ψb, ii If G is SO 3 then τ G = 1. Thus we have that = fκ g h τ 1 = fκ g τ 1 T ψb, B = g τ 1 h T ψn. N g τ 1 iii If G is SU then τ G = 1. Thus we have that = fκ g hτ 1 N = fκ g hτ 1 T ψb, B = gτ 1 h T ψn. B, h B, gτ 1 h B, Theorem 3.5. Let the curve flow α = ft gn hb be inextensible in three dimensional Lie group G. Then, there exists the following system of partial differential equations: κ = gκ f κ g h τ τ G h τ τ G gτ τ G, κψ = τ τ G fκ g hτ τ G gτ τ G h, τ τ G = gκτ τ G κ h ψ.

8 3 Gökmen Yıldız, O. Zeki Okuyucu Proof. Since = f = = [ fκ g, we have hτ τ G κ f κ g h fκ g g gτ τ G h N gτ τ G h τ τ G h τ τ G hτ τ G κt τ τ G B τ τ G g τ τ G τ τ G N ] B N 3.5 h B 3.6 and = κ κn = N κ. 3.7 Therefore, we followed 3.5 and 3.7 that κ = gκ f κ g h τ τ G h τ τ G gτ τ G and κψ = τ τ G fκ g hτ τ G gτ τ G h. Since B and B = Hence = = B [, we obtain gτ τ G h g τ τ G g τ τ G gτ τ G h ] T ψn h T κn ψ N ψ κt τ τ GB B = τ τ GN = τ τ G N τ τ G. τ τ G = gκτ τ G κ h ψ. No other new formulas are obtained from the relation =.

9 Inextensible Flows of Curves in Lie Groups 31 Theorem 3.5 includes the following special cases for system of partial differential equations: i If G is Abelian group then τ G =. Thus we have that κ = gκ f κ g κψ = τ fκ g hτ h τ h τ gτ, gτ h, τ = gκτ κ h ψ. ii If G is SO 3 then τ G = 1. Thus we have that κ = gκ f κ g h fκ g τ 1 τ 1 h κψ = τ 1 hτ 1 τ 1 = gκ τ 1 h κ iii If G is SU then τ G = 1. Thus we have that κ = gκ f κ g κψ = τ 1 fκ g hτ 1 τ 1 g τ 1 ψ. g τ 1, h, h 1 τ 1 h τ gτ 1, gτ 1 h, = gκτ 1 κ h ψ. 4. CONCLUSION The above noted special cases are showed us, the study is a generalization of inextensible flows of curves defined by Kwon [8] in Eculidean 3-space. Also, the similar conditions for inextensible flows of curves can investigate using the Lie group structure in different spaces or on other manifolds in addition to this study. References [1] G. Chirikjian, J. Burdick, A modal approach to hyper-redundant manipulator kinematics, IEEE Trans. Robot. Autom , [] P. Crouch, F. Silva Leite, The dynamic interpolation problem: on Riemannian manifoldsi Lie groups and symmetric spaces, J. Dyn. Control Syst [3] Ü. Çiftçi, A generalization of Lancert s theorem, J. Geom. Phys

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