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On the Differential Geometric Elements of Mannheim Darboux Applied Mathematical Sciences, Vol. 10, 2016, no. 62, 3087 - 3094 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2016.67221 On the Differential Geometric Elements of Mannheim Darboux Ruled Surface in E3 S¸eyda Kılı¸co˘glu 1 Faculty of Education, Department of Mathematics Baskent University, Ankara, Turkey S¨uleyman S¸enyurt Faculty of Arts and Sciences, Department of Mathematics Ordu University, Ordu, Turkey Copyright c 2016 S¸eyda Kılı¸co˘gluand S¨uleymanS¸enyurt. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribu- tion, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we consider two special ruled surfaces associated to Mannheim pair fα; α∗g. First, Mannheim Darboux Ruled surface (MDRS) of the curve α be defined and examined in terms of the Frenet-Serret appa- ratus of the Mannheim curve α, in E3. Further we have examined the differential geometric elements such as, Weingarten map S, Gaus cur- vature K and mean curvature H of Darboux ruled surface (DRS) and Mannheim Darboux ruled surface (MDRS) relative to each other. Also the first, second and third fundamental forms of Mannheim Darboux ruled surface (MDRS) have been examined in terms of the Mannheim curve α too. Mathematics Subject Classification: 53A04, 53A05 Keywords: Ruled surface, Darboux vector, Mannheim curves 1 Introduction and Preliminaries Involute-evolute curves, Bertrand curves, and Mannheim partner curves are the curves derivyed based on the other curves in geometry. Mannheim curve 1Corresponding author 3088 S¸eyda Kılı¸co˘gluand S¨uleymanS¸enyurt was firstly defined by A. Mannheim in 1878. A curve is called a Mannheim k1 curve if and only if 2 2 is a nonzero constant, with curvatures k1 and k2. k1+k2 Recently, a new definition of the associated curves was given by Liu and Wang [4]. According to this new definition, if the principal normal vector of first curve and binormal vector of second curve are linearly dependent, then first curve is called Mannheim curve, and the second curve is called Mannheim partner curve. As a result they called these new curves as Mannheim pair curves. For more detail see in [3] and [4]. The quantities fV1;V2;V3; D; k1; k2g are collectively Frenet-Serret apparatus of the curve α : I ! E3: Also 2 _ 3 2 3 2 3 V1 0 k1 0 V1 _ 4 V2 5 = 4 −k1 0 k2 5 4 V2 5 : (1.1) _ V3 0 −k2 0 V3 are well known the Frenet formulae. Darboux vector D is the areal velocity vector of the Frenet frame of a space curve. For any unit speed curve α;in terms of the Frenet-Serret apparatus, the Darboux vector can be expressed as D(s) = k2(s)V1 (s) + k1(s)V3 (s) (1.2) where curvature functions are k1 and k2 [1]. Along curve α under the condition that k1 6= 0; vector field ~ k2 D(s) = (s)V1 (s) + V3 (s) (1.3) k1 is called the modified Darboux vector field of curve α in [2]. Let α : I ! E3 and α∗ : I ! E3 be the C2 −class differentiable unit speed and ∗ 3 ∗ ∗ ∗ ∗ ∗ ∗ α : I ! E be two curves and let V1 (s) ;V2 (s) ;V3 (s) and V1 (s ) ;V2 (s ) ;V3 (s ) be the Frenet frames of the curves α and α∗, respectively. If the principal nor- ∗ mal vector V2 of the curve α is linearly dependent on the binormal vector V3 of the curve α∗, then the pair fα; α∗g is said to be Mannheim pair, then α is called a Mannheim curve and α∗ is called Mannheim partner curve of α where ∗ k1 < (V1;V1 ) = cos θ and besides the equality 2 2 = nonzero constant is known k1+k2 the offset property. In [5] and [6] Mannheim partner curves and Mannheim offsets of ruled surfaces are defined and characterized. Mannheim pair fα; α∗g can be represented by ∗ ∗ ∗ ∗ ∗ ∗ α(s ) = α (s ) + λ(s )V3 (s ) (1.4) for some function λ, since V2 and V3 are linearly dependent. This equation can be rewritten as ∗ α (s) = α (s) − λV2 (s) (1.5) Differential geometric elements of Mannheim Darboux ruled surface 3089 −k1 ∗ where λ = 2 2 : Frenet-Serret apparatus of Mannheim partner curve α , k1+k2 based in Frenet-Serret vectors of Mannheim curve α are ∗ V1 = cosθ V1 − sinθ V3 ∗ V2 = sinθ V1 + cosθ V3 (1.6) ∗ V3 = V2: The curvature and the torsion have the following equalyties, dθ θ_ k∗ = − = ; (1.7) 1 ds∗ cos θ ∗ k2 = k1sinθcosθ − k2cosθcosθ or ∗ k1 k2 = : (1.8) λk2 We use dot to denote the derivative with respect to the arc length parameter of the curve α: Also ds 1 −λk∗ = = 2 ; (1.9) ds∗ cos θ sinθ where jλj is the distance between the curves α and α∗; since d (α (s) ; α∗ (s)) = jλj. For more detail see in [5]. Also we can write ds 1 p ∗ = : (1.10) ds 1 + λk2 The product of Frenet vector fields of the Mannheim pair fα; α∗g has the following matrix form; 2 ∗ 3 2 3 V1 cosθ 0 −sinθ ∗ 4 V2 5 V1 V2 V3 = 4 sinθ 0 cosθ 5 : (1.11) ∗ V3 0 1 0 Definition 1.1 Ruled surface is said to be \ Darboux Ruled surface" if it is generated by moving Darboux vector fields, with the parametrization ~ k2 ' (s; u) = α (s) + uD(s)v = α (s) + u (s)V1 (s) + uV3 (s) : (1.12) k1 Also it has been called rectifying developable surface in [2]. 2 Mannheim Darboux Ruled surface In this section we will define and work on MDRS; which is known as rectifying developable ruled surface, or D~ scroll as in [7] where the differential geometric elements of the involute D~ − scroll:are examined too. Here first we will give Darboux vector field of the Mannheim partner α∗ as in the following theorem. 3090 S¸eyda Kılı¸co˘gluand S¨uleymanS¸enyurt Theorem 2.1 The modified Darboux vector of Mannheim partner curve α∗of a Mannheim curve α; based on the Frenet apparatus of Mannheim curve α is 2 2 2 2 ~ ∗ k1 + k2 2 k1 + k2 D = − cos θV1 + V2 + cos θsinθ V3: (2.1) _ _ θk2 θk2 Proof. The desired result is obtained from equations (1.3), (1.6) and (1.8). Definition 2.2 The parametrization of MDRS; in terms of the Frenet-Serret apparatus of the Mannheim partner curve α is 2 2 2 2 ∗ k1 + k2 2 k1 + k2 ' (s; v) = α − v cos θ V1 + (v − λ) V2 + v cos θ sin θ V3: (2.2) _ _ θk2 θk2 since '∗(s; v) = α∗(s) + vD~ ∗(s); where 2 2 2 2 ~ ∗ k1 + k2 2 k1 + k2 D = − cos θV1 + V2 + cos θ sin θ V3: _ _ θk2 θk2 Theorem 2.3 DRS and MDRS are intersect each other along the curve 2 ∗ sin θ cos θ sin θ ' (s) = α − V1 + V3: (2.3) θ_ θ_ Proof. With the equations 2 ∗ k1 cos θ k1 cos θ sin θ ' (s; v) = α + v V1 + (v − λ) V2 − v V3 λk2 θ_ λk2 θ_ and k2 ' (s; u) = α + u V1 + uV3 k1 under the conditions, 2 8 v k1 cos θ = u k2 < λk2 θ_ k1 v − λ = 0 : −v k1 cos θsinθ = u; λk2 θ_ 2 8 k1 cos θ = u k2 < k2 θ_ k1 v = λ : − k1 cos θsinθ = u; k2 θ_ we have k 1 = − tan θ: k2 This complete the proof. Differential geometric elements of Mannheim Darboux ruled surface 3091 Theorem 2.4 Normal vector field N of DRS is pependicular to the normal vector field N ∗of MDRS. Proof. Since the normal vector field N of DR α is 'sΛ'u N = = V2 k'sΛ'uk [7], and the normal vector field N ∗ of MDRS of the curve α is ∗ ∗ ∗ 'sΛ'u ∗ N = ∗ ∗ = V2 = sinθ V1 + cosθ V3: k'sΛ'uk ∗ it is trivial that hV2;V2 i = 0. Theorem 2.5 The matrix corresponding to the Weingarten map (Shape Op- erator ) S∗ of MDRS is 2 θ_ 3 − cos θ 0 0 k2+k2 S∗ = 6 1+ v 1 2 θ_ cos θ+ k1 (−θ_ sin θ+θ¨cos θ) 7 : (2.4) 4 θ_2 cos θ −k2 λk2 5 0 0 Proof. In the Euclidean 3−space; the matrix corresponding to the Weingarten map (Shape Operator ) S of DRS of curve α is " −k1 # 0 0 1+u k2 S = k1 : 0 0 [7]. Hence the matrix corresponding to the Weingarten map (Shape Operator ) S∗ of MDRS is ∗ 2 −k1 3 0 0 k∗ ∗ 1+v 2 S = k∗ 4 1 s∗ 5 0 0 ∗ ∗ ∗ by substituing k2 and k1 in matrix S we get 2 θ_ 3 − cos θ 0 10 0 k1 ∗ 6 1+v λk2 7 S = 6 @ θ_ A 7 cos θ 4 s∗ 5 0 0 Since the derivative with respect to parameter s∗ is 0 d k1 cos θ k cos θ λk2 θ_ ds 1 = _ ∗ λk2 θ s∗ ds ds 1 k2 + k2 0 k2 + k2 = 1 2 θ_ cos θ + 1 2 −θ_ sin θ + θ¨cos θ : θ_2 cos θ −k2 −k2 k2+k2 we have the proof. Where k1 = 1 2 : λk2 −k2 3092 S¸eyda Kılı¸co˘gluand S¨uleymanS¸enyurt Corollary 2.1 The Gaussian curvature of MDRS is K = det S∗ = 0: (2.5) Corollary 2.2 The mean curvature of MDRS is −k∗ H = traceS∗ = 1 ; (2.6) ∗ 0 k2 1 + u ∗ k1 − θ_ H = cos θ : 0 k2+k2 1 + v 1 2 θ_ cos θ + k1 −θ_ sin θ + θ¨cos θ θ_2 cos θ −k2 λk2 ∗ where < (V1;V1 ) = cos θ and θ is not constant.
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