Classifications of Special Curves in the Three-Dimensional Lie Group

International Journal of Mathematical Analysis Vol. 10, 2016, no. 11, 503-514 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2016.6230 Classifications of Special Curves in the Three-Dimensional Lie Group Chul Woo Lee Department of Mathematics Kyungpook National University Daegu 702-701, Republic of Korea Jae Won Lee Department of Mathematics Education and RINS Gyeongsang National University Jinju 660-701, Republic of Korea Copyright c 2016 Chul Woo Lee and Jae Won Lee. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In the present paper, we define a Bertrand curve in the three-dimensional Lie group G with a bi-invariant metric, and we show a Frenet curve α with Frenet curvatures k 1 and k 2 in G is a Bertrand curve if and only if it satisfies Ak 1 + B(k 2 + k 2 ) = 1, where A and B are some constants and k 2 = 1/2 [V 1, V 2 ], V 3. Also, we investigate a Bertrand curve using the Frenet curvature conditions of AW(k)-type (k = 1, 2, 3) curves in G. Mathematics Subject Classification: 53B25, 22E15 Keywords: Lie group, AW(k)-type curve, Bertrand curve, slant helix Correspondence author

504 Chul Woo Lee and Jae Won Lee 1 Introduction The geometry of curves and surfaces in the three-dimensional Euclidean space R 3 represented a popular topic in the field of classical differential geometry for many years. An increasing interest of the theory of curves makes a development of special curves to be examined. A way to the characterizations and the classifications for curves is the relationship between the Frenet curvatures of the curves. One of the curves is a Bertrand curve. Two curves which, at any point, have a common principal normal vector are Bertrand curves. The notion of Bertrand curves was discovered by J. Bertrand in 1850. Bertrand curves have attracted many mathematicians since the beginning. In [14], Pears extended the well-known properties of Bertrand curves in R 3 to curves in the n-dimensional Euclidean space R n. Also, many authors have studied Bertrand curves in other ambient spaces(endoed with Riemannian or pseudo-riemannian metric): in the three dimensional Lorentz-Minkowski space R 3 1 [3], [4], in Riemann-Otsuki space [16], in the Galilean space [13], in space forms with sectional curvature c [5], and so on. The notion of AW(k)-type submanifolds was introduced by Arslan and West in [2]. In particular, many works with AW(k)-type curves have been done by several authors. For example, in [9], the authors gave curvature conditions and charaterizations related to these curves in R n. Also, in [10], they investigated curves of AW(k)-type in the three-dimensional null cone and gave curvature conditions of these kind of curves. On the other hand, in [11] and [15], authors studied Bertrand curves of AW(k)-type in Lorentz-Minkowski space. However, to the author s knowledge, there is no article dedicated to studying the notions of Bertrand curves and AW(k)-type curves immersed in Lie group. The main goal of this paper is to define Bertrand curves in the three dimensional Lie group G with a bi-invariant metric. Also, we investigate Bertrand curves using the Frenet curvature conditions of AW(k)-type curves in G. 2 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 identity of G. On the other hand, the following equations with respect to bi-invariant metric are satisfied [7] X, [Y, Z] = [X, Y ], Z (2.1) and D X Y = 1 [X, Y ] (2.2) 2

Classifications of special curves in the three-dimensional Lie group 505 for all X, Y, Z g. Let α : I R G be a unit speed curve with a parameter s and {V 1, V 2,, V n } be an orthonrmal basis of g. In this case, we write that any vector fields W and Z along the curve α as W = n i=1 w iv i and Z = n i=1 z iv i, where w i : I R and z i : I R are smooth functions. Furthermore, the Lie bracket of two vector fields W and Z is given by [W, Z] = n w i z j [V i, V j ]. (2.3) i,j=1 Let D α W be the covariant derivative of W along the curve α, V 1 = α and W = n i=1 w iv i, where w i = dw i /ds. Then we have D α W = W + 1 2 [V 1, W ]. (2.4) A curve α is called a Frenet curve of osculating order d if its derivatives α (s), α (s), α (s),...,α (d) (s) are linearly dependent and α (s), α (s), α (s),...,α (d+1) (s) are no longer linearly independent for all s. To each Frenet curve of order d one can associate an orthonormal d-frame V 1 (s), V 2 (s), V 3 (s),...,v d (s) along α (such that α (s) = V 1 (s)) called the Frenet frame and the functions k 1, k 2,..., k d 1 : I R said to be the Frenet curvatures such that the Frenet formulas are defined in the usual way: D V1 V 1 (s) = k 1 (s)v 2 (s), D V1 V 2 (s) = k 1 (s)v 1 (s) + k 2 (s)v 3 (s), D V1 V i (s) = k i 1 (s)v i 1 (s) + k i (s)v i+1 (s), D V1 V i+1 (s) = k i (s)v i (s). Let α : I G be a Frenet curve of osculating order 3 in Lie group G with a bi-invariant metric. We define k 2 (s) = 1 2 [V 1, V 2 ], V 3. (2.5) Proposition 2.1 ([17]). Let α be a Frenet curve of osculating order 3 in Lie group G with a bi-invariant metric. Then we have [V 1, V 2 ] = [V 1, V 2 ], V 3 V 3 = 2 k 2 V 3, [V 1, V 3 ] = [V 1, V 3 ], V 2 V 2 = 2 k 2 V 2, [V 2, V 3 ] = [V 2, V 3 ], V 1 V 1 = 2 k 2 V 1.

506 Chul Woo Lee and Jae Won Lee Remark. Let G be the three-dimensional Lie group with a bi-invariant metric. Then it is one of the Lie groups SO(3), S 3 or a commutative group and the following statements hold (see [6], [8]): (i) If G is SO(3), then k 2 (s) = 1 2. (ii) If G is S 3 = SU(2), then k2 (s) = 1. (iii) If G is a commutative group, that is, S 1 S 1 S 1, S 1 R 2, S 1 S 1 R or R 3, then k 2 (s) = 0. Theorem 2.2 ([6]). A Frenet curve of osculating order 3 in Lie group G with a bi-invariant metric is a general helix if and only if where c is a constant. k 2 (s) = ck 1 (s) + k 2 (s), (2.6) From (2.6), a curve with k 1 0 is a general helix if and only if ( k 2 k 2 k 1 )(s)= constant. As a Euclidean sense, if both k 1 (s) 0 and k 2 (s) k 2 (s) are constants, it is a cylindrical helix. We call such a curve a circular helix. Theorem 2.3 ([12]). A Frenet curve of osculating order 3 in Lie group G with a bi-invariant metric is a slant helix if and only if is a constant, where H = k 2(s) k 2 (s) k 1 (s). k 1 (s)(1 + H 2 ) 3 2 H (2.7) Proposition 2.2 ([17]). Let α be a Frenet curve of osculating order 3 in Lie group G with a bi-invariant metric. Then we have α (s) = V 1 (s), α (s) = k 1 (s)v 2 (s), α (s) = k 2 1(s)V 1 (s) + k 1(s)V 2 (s) + k 1 (s)(k 2 (s) k 2 (s))v 3 (s), α (s) = 3k 1 (s)k 1(s)V 1 (s) + [k 1 (s) k 3 1(s) k 1 (s)k 2 2(s) + 2k 1 (s)k 2 (s) k 2 (s) k 1 (s) k 2 2 (s)]v2 (s) + [2k 1(s)(k 2 (s) k 2 (s)) + k 1 (s)(k 2 (s) k 2 (s)) ]V 3 (s). Notation. Let we put N 1 (s) = k 1 (s)v 2 (s), N 2 (s) = k 1(s)V 2 (s) + k 1 (s)(k 2 (s) k 2 (s))v 3 (s), N 3 (s) = [k 1(s) k 3 1(s) k 1 (s)k 2 2(s) + 2k 1 (s)k 2 (s) k 2 (s) k 1 (s) k 2 2 (s)]v 2 (s) + [2k 1(s)(k 2 (s) k 2 (s)) + k 1 (s)(k 2 (s) k 2 (s)) ]V 3 (s).

Classifications of special curves in the three-dimensional Lie group 507 3 Curves of AW(k)-Type In this section, we consider the properties of curves of AW(k)-type in Lie group G. Definition 3.1 (cf. [1]). Frenet curves of osculating order 3 in Lie group G with a bi-invariant metric are (i) of weak AW(2)-type if they satisfy (ii) of weak AW(3)-type if they satisfy N 3 (s) = N 3 (s), N 2 (s) N 2 (s), (3.1) N 3 (s) = N 3 (s), N 1 (s) N 1 (s), (3.2) where N 1 (s) = N 1(s) N 1 (s), N2 (s) = N 2(s) N 2 (s), N1 (s) N1 (s). N 2 (s), N1 (s) N1 (s) Definition 3.2 ([2]). Frenet curves of osculating order 3 in Lie group G with a bi-invariant metric are (i) of AW(1)-type if they satisfy N 3 (s) = 0, (ii) of AW(2)-type if they satisfy N 2 (s) 2 N 3 (s) = N 3 (s), N 2 (s) N 2 (s), (3.3) (iii) of AW(3)-type if they satisfy N 1 (s) 2 N 3 (s) = N 3 (s), N 1 (s) N 1 (s). (3.4) From the definitions of AW(k)-type, we have the following propositions ([17]): Proposition 3.3. Let α be a Frenet curve of osculating order 3 in Lie group G with a bi-invariant metric. Then α is of weak AW (2)-type if and only if k 1(s) k 3 1(s) k 1 (s)k 2 2(s) + 2k 1 (s)k 2 (s) k 2 (s) k 1 (s) k 2 2 (s) = 0. (3.5) Proposition 3.4. Let α be a Frenet curve of osculating order 3 in Lie group G with a bi-invariant metric. Then α is of weak AW (3)-type if and only if 2k 1(s)(k 2 (s) k 2 (s)) + k 1 (s)(k 2 (s) k 2 (s)) = 0. (3.6)

508 Chul Woo Lee and Jae Won Lee Proposition 3.5. Let α be a Frenet curve of osculating order 3 in Lie group G with a bi-invariant metric. Then α is of AW (1)-type if and only if k 1(s) k 3 1(s) k 1 (s)k 2 2(s) + 2k 1 (s)k 2 (s) k 2 (s) k 1 (s) k 2 2 (s) = 0, k 2 1(s)(k 2 (s) k 2 (s)) = c, (3.7) where c is a constant. Proposition 3.6. Let α be a Frenet curve of osculating order 3 in Lie group G with a bi-invariant metric. Then α is of AW (2)-type if and only if k 1(s)[2k 1(s)(k 2 (s) k 2 (s)) + k 1 (s)(k 2 (s) k 2 (s)) ] = k 1 (s)(k 2 (s) k 2 (s))[k 1(s) k 3 1(s) k 1 (s)k 2 2(s) + 2k 1 (s)k 2 (s) k 2 (s) k 1 (s) k 2 2 (s)] = 0. (3.8) Proposition 3.7. Let α be a Frenet curve of osculating order 3 in Lie group G with a bi-invariant metric. Then α is of AW (3)-type if and only if where c is a constant. k 2 1(s)(k 2 (s) k 2 (s)) = c, (3.9) 4 Bertrand Curves in Lie Group In this section, we define a Bertrand curve in the three-dimensional Lie group G and investigate the properties of it. Definition 4.1. A curve α : I G with k 1 (s) 0 in the three-dimensional Lie group G is called a Bertrand curve if there exists a curve β : I G such that, at the corresponding points of curves, the principal normal lines of α(s) and β( s) on s, s I are equal. In this case, β( s) is called a Bertrand mate of α(s). Let W 1, W 2, W 3 be the Frenet frame of the Bertrand mate β( s) of α(s) and l 1, l 2 be the Frenet curvatures of β( s). Theorem 4.2. Let β be a Bertrand mate of a Bertrand curve α in the threedimensional Lie group G with a bi-invariant metric. Then there exists a constant λ such that β( s) = α(s) + λv 2 (s). (4.1)

Classifications of special curves in the three-dimensional Lie group 509 Proof. Let β( s) be the Bertrand mate of a Bertrand curve α(s). Then β( s) can be expressed as β( s) = α(s) + λ(s)v 2 (s), (4.2) for some smooth function λ(s) 0. Differentiating (4.2) with respect to s, we have W 1 ( s) d s ds = V 1(s) + λ (s)v 2 (s) + λ(s)(d α V 2 (s) 1 2 [V 1(s), V 2 (s)]) = V 1 (s)) + λ (s)v 2 (s) + λ(s)( k 1 (s)v 1 (s) + k 2 (s)v 3 (s) k 2 (s)v 3 (s)]) = (1 λ(s)k 1 (s))v 1 (s) + λ (s)v 2 (s) + λ(s)(k 2 (s) k 2 (s))v 3 (s). (4.3) Since W 1 ( s), V 2 (s) = 0, we get λ (s) = 0, that is, λ(s) is a constant. Theorem 4.3. If β is the Bertrand mate of α in the three-dimensional Lie group G with a bi-invariant metric, then the angle between the tangent vectors at the corresponding points of α and β is a constant. Proof. If we show that d ds V 1(s), W 1 ( s) = 0, then the proof is complete. d ds V 1(s), W 1 ( s) = dv 1(s), W 1 ( s) + d s ds ds V 1(s), dw 1( s) d s = D α (s)v 1 (s) 1 2 [V 1(s), V 1 (s)], W 1 ( s) + d s ds V 1(s), D β ( s)w 1 ( s) 1 2 [W 1( s), W 1 ( s)] = k 1 (s) V 2 (s), W 1 ( s) + l 1 ( s) d s ds V 1(s), W 2 ( s) = 0 (4.4) because V 2 (s), W 1 ( s) = 0 and V 1 (s), W 2 ( s) = 0. Theorem 4.4. A curve α with k 2 (s) 0 in the three-dimensional Lie group G with a bi-invariant metric is a Bertrand curve if and only if there exist non-zero constants A, B such that Ak 1 (s) + B(k 2 (s) k 2 (s)) = 1. (4.5) Proof. From (4.5), we have V 1 (s), W 1 ( s) = cos θ = constant. It follows that we can obtain V 3 (s), W 3 ( s) = cos θ.

510 Chul Woo Lee and Jae Won Lee In fact, V 3 (s), W 3 ( s) = V 1 (s) V 2 (s), W 1 ( s) W 2 ( s) = V 1 (s), W 1 ( s) V 2 (s), W 2 ( s) V 1 (s), W 2 ( s) V 2 (s), W 1 ( s) = V 1 (s), W 1 ( s) = cos θ. From this, V 1 (s), W 3 ( s) = cos(90 o θ) = sin θ. nonzero constant in (4.3), we have Considering that λ is a W 1 ( s) d s ds = (1 λk 1(s))V 1 (s) + λ(k 2 (s) k 2 (s))v 3 (s), (4.6) it follows that we obtain 0 = (1 λk 1 (s)) V 1 (s), W 3 ( s) + λ(k 2 (s) k 2 (s)) V 3 (s), W 3 ( s) = (1 λk 1 (s)) sin θ + λ(k 2 (s) k 2 (s)) cos θ. Thus, we have the required linear relation (4.5) between k 1 (s) and k 2 (s) k 2 (s) for α. Conversely, let A = λ in (4.5), and define β( s) = α(s) + λv 2 (s). Then, by again using the relation (4.5), (4.6) becomes W 1 ( s) d s ds = (k 2(s) k 2 (s))(bv 1 (s) + λv 3 (s)). Thus, a unit vector W 1 ( s) of β( s) is given by W 1 ( s) = 1 B 2 +λ 2 (BV 1 (s)+λv 3 (s)). It follows that dw 1 ( s) ds 1 = B2 + λ [Bk 1(s) + λ(k 2 (s) k 2 (s))]v 2 (s). 2 Therefore, V 2 (s) = ±W 2 ( s), and the principal normal lines of α and β agree. Thus, α is a Bertrand curve. Corollary 4.5. A circular helix in the three-dimensional Lie group G with a bi-invariant metric is a Bertrand curve. Proposition 4.6. A Frenet curve α with k 1 (s) 0 and k 2 (s) k 2 (s) 0 in the three-dimensional Lie group G with a bi-invariant metric is a Bertrand curve if and only if there is a non-zero constant A such that A[k 1 (s)(k 2 (s) k 2 (s)) k 1(s)(k 2 (s) k 2 (s))] (k 2 (s) k 2 (s)) = 0. (4.7) Proof. From (4.5), we have 1 Ak 1(s) k 2 (s) k 2 = B. If we differentiate both sides of the (s) last equation, we get (4.7). It can be easily shown that the converse assertion is also true.

Classifications of special curves in the three-dimensional Lie group 511 5 Bertrand Curves and Slant Helix of AW(k)- Type In this section, we study Bertrand curves and slant helices of AW(k)-type in the three-dimensional Lie group G with a bi-invariant metric and characterize these curves. Theorem 5.1. Let α be a Bertrand curve with k 1 (s) 0 and k 2 (s) k 2 (s) 0 in the three-dimensional Lie group G with a bi-invariant metric. Then α is of AW(3)-type if and only if it is a circular helix. Proof. Suppose that α is a Bertrand curve of AW(3)-type. Then (3.9) and (4.7) are satisfied. By taking differentiation both sides of (3.9) with respect to s, (k 2 (s) k 2 (s)) = 2ck 1(s) k 3 1(s). (5.1) If we substitute (3.9) and (5.1) in (4.7), we get it follows that from (3.9) we have k 1 (s) = 2 3A, (5.2) k 2 (s) k 2 (s) = 9cA2 4. (5.3) Thus α is a circular helix. The converse assertion is trivial. Hence the theorem is proved. Theorem 5.2. There is no any Bertrand curve with k 1 (s) 0 and k 2 (s) k 2 (s) 0 of AW(1)-type in the three-dimensional Lie group G with a biinvariant metric. Proof. Suppose that α is a Bertrand curve of AW(1)-type. Then (3.7) and (4.7) hold on α. Combining the second equation in (3.7) and (4.7), we easily seen that k 1 (s) = 2 3A. (5.4) If we substitute (5.4) in (4.7), one find which implies 8 27A + 27c2 A 3 3 8 A 6 = 64 (27c) 2. = 0,

512 Chul Woo Lee and Jae Won Lee It is impossible. So, α cannot be a AW(1)-type Bertrand curve. Hence the theorem is proved. Theorem 5.3. If α is of weak AW(3)-type with k 1 (s) 0 and k 2 (s) k 2 (s) 0 in the three-dimensional Lie group G with a bi-invariant metric, then it is of AW(3)-type. Proof. Suppose that α is of weak AW(3)-type. Then the following equation is satisfied From this, we obtain 2k 1(s)(k 2 (s) k 2 (s)) + k 1 (s)(k 2 (s) k 2 (s)) = 0. k 2 1(s)(k 2 (s) k 2 (s)) = c 1, where c 1 is a constant. Thus, by proposition 3.7 α is of AW(3)-type. Combining Theorems 5.1 and 5.3, we have Theorem 5.4. Let α be a Bertrand curve with k 1 (s) 0 and k 2 (s) k 2 (s) 0 in the three-dimensional Lie group G with a bi-invariant metric. If α is of weak AW(3)-type, then it is a circular helix. From (2.7) one can get easily the following proposition: Proposition 5.5. Let α be a curve with k 1 (s) 0 in the three-dimensional Lie group G with a bi-invariant metric. Then α is a slant helix if and only if there is a constant c 1 such that (k 2 1(s) + (k 2 (s) k 2 (s)) 2 ) 3 2 = c1 [k 1 (s)(k 2 (s) k 2 (s)) k 1(s)(k 2 (s) k 2 (s))]. (5.5) Theorem 5.6. Let α be a Bertrand curve with k 1 (s) 0 and k 2 (s) k 2 (s) 0 in the three-dimensional Lie group G with a bi-invariant metric. If there is a non-zero constant c 2 such that then α is a slant helix. (k 2 (s) k 2 (s)) = c 2 [k 2 1(s) + (k 2 (s) k 2 (s)) 2 ] 3 2, (5.6) Proof. Since α be a Bertrand curve with k 1 (s) 0 and k 2 (s) k 2 (s) 0, (4.7) holds, that is, k 1 (s)(k 2 (s) k 2 (s)) k 1(s)(k 2 (s) k 2 (s)) = 1 A (k 2(s) k 2 (s)). (5.7) If (5.6) is satisfied, then from (5.5) and (5.7) it is easy to that α is a slant helix.

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