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OSCULATING CURVES in the GALILEAN 4-SPACE Dae Won Yoon1, Jae Won Lee2, Chul Woo Lee3 § 1Department of Mathematics Education

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OSCULATING CURVES in the GALILEAN 4-SPACE Dae Won Yoon1, Jae Won Lee2, Chul Woo Lee3 § 1Department of Mathematics Education International Journal of Pure and Applied Mathematics Volume 100 No. 4 2015, 497-506 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v100i4.9 ijpam.eu OSCULATING CURVES IN THE GALILEAN 4-SPACE Dae Won Yoon1, Jae Won Lee2, Chul Woo Lee3 § 1Department of Mathematics Education and RINS Gyeongsang National University Jinju, 660-701, REPUBLIC OF KOREA 2Department of Mathematics Education Busan National University of Education Busan 611-736, REPUBLIC OF KOREA 3Department of Mathematics Kyungpook National University Daegu, 702-701, REPUBLIC OF KOREA Abstract: In this paper, we study osculating curves and equiform osculating curves in the 4-dimensional Galilean space G4 and characterize such curves in terms of their curvature functions and their equiform curvature functions, respectively. AMS Subject Classification: 53A35, 53C44 Key Words: Galilean space, equiform geometry, osculating curve 1. Introduction In the Euclidean space E3, there exist three classes of curves, called rectifying, normal, and osculating curves, which satisfy the Cesaro’s fixed point condi- tion (see [5]). Namely, rectifying, normal, and osculating planes of such curves always contain a particular point. It is well-known that if all the normal or osculating planes of a curve in E3 pass through a particular point, then the c 2015 Academic Publications, Ltd. Received: January 5, 2015 url: www.acadpubl.eu §Correspondence author 498 D.W. Yoon, J.W. Lee, C.W. Lee curve lies in a sphere or is a planner curve, respectively. It is also known that if all rectifying planes of a non-planar curve in E3 pass through a particular point, then the ratio of torsion and curvature of such curve is a non-constant linear function (see [2]). Moreover, Ilarslan˙ and Neˇsovi˙c(see [4]) gave some charac- terizations for osculating curves in E3, and they also constructed osculating curves in E4 as a curve whose position vector all the time lies in the orthogonal complement of its first binormal vector field. As the results, they classified the osculating curves in terms of their curvature functions and gave the necessary and the sufficient conditions of osculating curves for arbitrary curves in E4. On the other hand, the equiform geometry of the Cayley-Klein space is defined by requesting that similarity group of the space preserves angles between planes and lines. Cayley-Klein geometries are studied for many years. However, they recently have become interesting again because of their importance for other fields, like soliton theory (see [7]) A Galilean space is one of the Cayley-Klein spaces and it has been largely developed by R¨oschel (see [6]). A Galilean space may be considered as the limit case of a pseudo-Euclidean space in which the isotropic cone degenerates to a plane. The limit transition corresponds to the limit transition from the special relatively theory to classical mechanics. In this paper, we study osculating curves and equiform osculating curves in the 4-dimensional Galilean space G4 and characterize such curves in terms of their curvature functions and their equiform curvature functions, respectively. 2. Preliminaries The 3-dimensional Galilean space G3 is the Cayley-Klein space equipped with the projective metric of signature (0, 0, +, +). The absolute figure of the Galilean space consists of an ordered triple {w, f, I}, where w is the ideal (absolute) plane, f is the line (absolute line) in w and I is the fixed elliptic involution of points of f. The study of mechanics of plane-parallel motions reduces to the study of a geometry of the 3-dimensional space with coordinates {x, y, t}, given by the motion formula. It is explained that the 4-dimensional Galilean geometry, which studies all properties invariant under motions of objects in the space, is even complex. In addition, it is started that this geometry can be described more precisely as the study of those properties of the 4-dimensional space with coordinates OSCULATING CURVES IN THE GALILEAN 4-SPACE 499 which are invariant under the general Galilean transformations as follows [8]: x′ =(cos β cos α − cos γ sin β sin α)x + (sin β cos α − cos γ cos β sin α)y + (sin γ sin α)z + (v cos δ1)t + a, y′ = − (cos β sin α + cos γ sin β cos α)x + (− sin β sin α − cos γ cos β cos α)y + (sin γ cos α)z + (v cos δ2)t + b, ′ z =(sin γ sin β)x − (sin γ cos β)y + (cos γ)z + (v cos δ3)t + c, t′ =t + d, 2 2 2 where cos δ1 + cos δ2 + cos δ3 = 1. The Galilean scalar product in G4 can be written as x y , if x 6= 0 or y 6= 0 hx, yi = 1 1 1 1 (2.1) (x2y2 + x3y3 + x4y4, if x1 = 0 and y1 = 0, where x = (x1, x2, x3, x4) and y = (y1, y2, y3, y4) are vectors in G4. It leaves invariant the Galilean norm of the vector x, defined by |x |, if x 6= 0 ||x|| = 1 1 2 2 2 ( x2 + x3 + x4, if x1 = 0. p The Galilean cross product of x, y and z on G4 is defined by 0 e2 e3 e4 x x x x x × y × z = 1 2 3 4 , y y y y 1 2 3 4 z z z z 1 2 3 4 where e2 = (0, 1, 0, 0), e3 = (0, 0, 1, 0), and e4 = (0, 0, 0, 1). ∞ A curve α : I ⊂ R → G4 of the class C in the Galilean space G4 is defined by the parametrization α(s) = (s, y(s), z(s), w(s)), where s is a Galilean invariant arc-length of α. 500 D.W. Yoon, J.W. Lee, C.W. Lee On the other hand, the Frenet vectors of α(s) in G4 are defined by dα(s) t(s) = =α ˙ (s) = (1, y˙(s), z˙(s), w˙ (s)) , ds 1 1 n(s) = α¨(s) = (0, y¨(s), z¨(s), w¨(s)) , κ1(s) κ1(s) d 1 y(¨s) d 1 z(¨s) d 1 w(¨s) 1 κ1(s) κ1(s) κ1(s) b1(s) = 0, , , , κ2(s) ds ds ds b2(s) = t(s) × n(s) × b1(s), where κ1(s), κ2(s), and κ3(s) are the first, second and third curvature functions, respectively, given by 2 2 2 κ1(s) = y¨(s) +z ¨(s) +w ¨(s) , q κ2(s) = hn˙ (s), n˙ (s)i, ˙ κ3(s) = phb1(s), b2(s)i. The vectors t, n, b1, b2 are called the tangent, principal normal, first binormal, and second binormal vectors of α, respectively. If the curvature functions κ1, κ2 and κ3 of α are constants, then a curve α is called a W -curve. For their derivatives the following Frenet formula satisfies (cf. [3]) t˙(s) = κ1(s)n(s), n˙ (s) = κ2(s)b1(s), (2.2) b˙1(s) = −κ2(s)n(s) + κ3(s)b2(s), b˙2(s) = −κ3(s)b1(s). Now, we define osculating curves in the Galilean space G4. Let α be a unit speed curve in G4. If its position vector always lies in the ⊥ ⊥ orthogonal complement b1 or b2 of b1 or b2, then a curve α is called an osculating curve in G4. Consequently, an osculating curve can be expressed as α(s) = λ(s)t(s) + µ(s)n(s) + ν(s)b2(s), (2.3) or α(s) = λ(s)t(s) + µ(s)n(s) + ν(s)b1(s) (2.4) for some smooth functions λ(s), µ(s) and ν(s). In this paper, we deal with an osculating curve generating by the tangent vector, the principal normal vector and the second binormal vector of the curve α in G4. OSCULATING CURVES IN THE GALILEAN 4-SPACE 501 3. Osculating Curves in G4 In this section, we characterize osculating curves in G4 in terms of their curva- tures. Theorem 1. Let α be a unit speed curve in G4 with non-zero curvatures κ1, κ2 and κ3. Then α is an osculating curve if and only if d κ˙ − 3 = s + c, (3.1) κ κ 1 2 where c, d 6= 0 are constant. Proof. Let α = α(s) be a unit speed osculating curve and κ1(s), κ2(s), and κ3(s) be non-zero curvatures of α. Then, the position vector α(s) of the curve α satisfies the following equation: α(s) = λ(s)t(s) + µ(s)n(s) + ν(s)b2(s) for some smooth functions λ(s), µ(s) and ν(s). Differentiating the above equa- tion with respect to s with the Frenet formulae (2.2), we obtain t = λ˙ t + (κ1λ +µ ˙ ) n + (κ2µ − νκ3) b1 +ν ˙b2. From this, we get λ˙ = 1, κ1λ +µ ˙ = 0, (3.2) κ2µ − νκ3 = 0, ν˙ = 0. It follows that λ = s + c µ κ3 d (3.3) = κ2 ν = d, c, d 6= 0 ∈ R. Using the second equation in (3.2) and (3.3), we obtain the curvatures κ1, κ2 and κ3 satisfying the equation d κ˙ − 3 = s + c. (3.4) κ κ 1 2 502 D.W. Yoon, J.W. Lee, C.W. Lee Conversely, assume that the curvatures κ1(s), κ2(s) and κ3(s) of a unit speed curve α in G4 satisfy equation (3.4). Let us consider the vector x ∈ G4, given by dκ3 x(s) = α(s) − (s + c) t(s) − n(s) − db2. κ2 Then, we can easily find x˙ (s) = 0, that is, x is a constant vector.
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