on mannheim partner curves in dual lorentzian space
Transkript
on mannheim partner curves in dual lorentzian space
Hacettepe Journal of Mathematics and Statistics Volume 40 (5) (2011), 649 – 661 ON MANNHEIM PARTNER CURVES IN DUAL LORENTZIAN SPACE Sıddıka Özkaldı∗†‡, Kazım İlarslan§ and Yusuf Yaylı∗ Received 28 : 12 : 2009 : Accepted 09 : 03 : 2011 Abstract In this paper we define non-null Mannheim partner curves in three dimensional dual Lorentzian space D31 , and obtain necessary and sufficient conditions for the existence of non-null Mannheim partner curves in dual Lorentzian space D31 . Keywords: curve. Mannheim partner curve, Dual Lorentzian space, Dual Lorentzian space 2000 AMS Classification: 53 A 25, 53 A 40, 53 B 30. 1. Introduction In the differential geometry of a regular curve in Euclidean 3-space E3 , it is well known that one of the important problems is the characterization of a regular curve. The curvature functions k1 (curvature κ) and k2 (torsion τ ) of a regular curve play an important role in determining the shape and size of the curve [4, 8]. For example: If k1 = k2 = 0, then the curve is a geodesic. If k1 6= 0 (constant) k2 = 0, then the curve is a circle with radius 1/k1 . If k1 6= 0 (constant) and k2 6= 0 (constant), then the curve is a helix in the space, etc. Another route to the classification and characterization of curves is the relationship between the Frenet vectors of the curves. For example Saint Venant, in 1845, proposed the question whether upon the surface generated by the principal normal of a curve, a second curve can exist which has for its principal normal the principal normal of the given curve. This question was answered by Bertrand in 1850, when he showed that a necessary and sufficient condition for the existence of such a second curve is that a linear ∗ Department of Mathematics, Faculty of Science, Ankara University, Tandoğan, Ankara, Turkey. E-mail: (S. Özkaldı) [email protected] (Y. Yaylı) [email protected] † Current Address: Department of Mathematics, Faculty of Science and Arts, Bilecik University, Bilecik, Turkey. E-mail: [email protected] ‡ Corresponding Author. § Department of Mathematics, Faculty of Science and Arts, Kırıkkale University, Kırıkkale, Turkey. E-mail: [email protected] 650 S. Özkaldı, K. İlarslan, Y. Yaylı relationship with constant coefficients shall exist between the first and second curvatures of the given original curve. The pairs of curves of this kind have been called Conjugate Bertrand Curves, or more commonly Bertrand Curves [4, 8, 13]. There are many works related with Bertrand curves in Euclidean space and Minkowski space. Associated curves of another kind, called Mannheim curves and Mannheim partner curves occur if there exists a relationship between the space curves α and β such that, at the corresponding points of the curves, the principal normal lines of α coincide with the binormal lines of β. Then, α is called a Mannheim curve, and β the Mannheim partner curve of α. Mannheim partner curves were studied by Liu and Wang [9] in Euclidean 3-space and in Minkowski 3-space. Dual numbers had been introduced by William Kingdon Clifford (1845–1879) as a tool for his geometrical investigations. After him Eduard Study (1862–1930) used dual numbers and dual vectors in his research on line geometry and kinematics. He devoted special attention to the representation of oriented lines by dual unit vectors and defined the famous mapping: The set of oriented lines in a Euclidean three-dimensional space E3 is one to one correspondence with the points of a dual space D3 of triples of dual numbers [5]. Differential Geometric properties of regular curves in a dual space D3 , as well as in a dual Lorentzian space D31 have been studied in many papers. For example see [16, 3, 7, 17, 18, 1, 2, 11, 15, 14]. Mannheim curves in a dual space were studied by the authors in [12] In this paper we study non-null Mannheim partner curves in a dual Lorentzian space D31 . 2. Preliminary Dual numbers were introduced by W. K. Clifford (1845–1879) as a tool for his geometrical investigations. After him, E. Study used dual numbers and dual vectors in his research on the geometry of lines and kinematics. He devoted special attention to the representation of directed lines by dual unit vectors, and defined the mapping that is known by his name. Namely, there exists one-to-one correspondence between the points of dual unit sphere S2 and the directed lines in R3 [5]. If we take the Minkowski 3-space R31 instead of R3 the E. Study (1862–1930) mapping can be stated as follows: The dual timelike and spacelike unit vectors of the dual hyperbolic and Lorentzian unit spheres H20 and S21 in the dual Lorentzian space D31 are in one-to-one correspondence with the directed timelike and spacelike lines in R31 , respectively. Then a differentiable curve on H20 corresponds to a timelike ruled surface at R31 . Similarly, the timelike (resp. spacelike) curve on S21 corresponds to any spacelike (resp. timelike) ruled surface in R31 . We will survey briefly various fundamental concepts and properties in Lorentzian space. We refer mainly to O’Neill [10]. Let R31 be the 3-dimensional Lorentzian space with Lorentzian metric h · , · i : −dx21 + dx22 +dx23 . It is known that in R31 there are three categories of curves and vectors, namely, → spacelike, timelike and null, depending on their causal character. Let − x be a tangent → − → − → − → − → vector of Lorentzian space. Then x is said to be spacelike if h x , x i > 0 or − x = 0, → − → → → → → timelike if h− x,− x i < 0, null (lightlike) if h− x,− x i = 0 and − x 6= 0 . Let α : I ⊂ R −→ R31 be a regular curve in R31 . Then, the curve α is spacelike if all its velocity vectors are spacelike. Similarly, it is called timelike and a null curve if all its velocity vectors are timelike and null vectors, respectively. On Mannheim Partner Curves in Dual Lorentzian Space 651 A dual number x b has the form x + εx∗ with the properties ε 6= 0, 0ε = ε0 = 0, 1ε = ε1 = ε, ε2 = 0, where x and x∗ are real numbers and ε is the dual unit (for the properties of dual numbers, see [16]). An ordered triple of dual numbers (b x1 , x b2 , x b3 ) is called a dual vector and the set of dual vectors is denoted by D3 = D × D × D − → − → = x b | x b = (x1 + εx∗1 , x2 + εx∗2 , x3 + εx∗3 ) → − − → ={x b | x b = (x1 , x2 , x3 ) + ε (x∗1 , x∗2 , x∗3 )} − − → → − → → − → − = x b | x b =→ x + εx ∗ , − x , x ∗ ∈ R3 − → − → − → − − → → For any x b =→ x + εx∗ , yb = − y + εy ∗ ∈ D3 , if the Lorentzian inner product of the dual → − → − vectors x b and yb is defined by − → − → − → − → → → − → x b , yb := − x,→ y +ε − x , y ∗ + x∗ , − y , then the dual space D3 together with this Lorentzian inner product is called the dual → − Lorentzian space, and it is shown by D31 . A dual vector x b in D3 is said to be spacelike, → − timelike and lightlike (null) if the vector x b is spacelike, timelike and lightlike (null), → − → − respectively. The Lorentzian vector product of dual vectors x b = (b x1 , x b2 , x b3 ) and x b = 3 (b x1 , x b2 , x b3 ) in D1 is defined by → − − → x b ∧ yb := (b x3 yb2 − x b2 yb3 , x b3 yb1 − x b1 yb3 , x b1 yb2 − x b2 yb1 ) . − − → → − − → → b =→ x + εx∗ is defined by If − x 6= 0, the norm x b of x r − → − → − → b, x b . b := x x − → − A dual vector → x with norm 1 is called a dual unit vector. Let x b = (b x1 , x b2 , x b3 ) ∈ D31 . Then, i) The set S21 = n− o − → − → → − → − − − x b =→ x + εx ∗ x b = (1, 0); → x , x∗ ∈ R31 and the vector → x is spacelike → − is called the pseudo dual sphere with center O in D31 . ii) The set H20 = n− o − → − → → − → − − − x b =→ x + εx ∗ x b = (1, 0); → x , x∗ ∈ R31 and the vector → x is timelike is called the pseudo dual hyperbolic space in D31 [15] If all the real valued functions x1 (t) and x∗1 , 1 ≤ i ≤ 3, are differentiable, the dual Lorentzian curve x b : I ⊂ R → D31 → − t 7→ x b (t) = (x1 (t) + εx∗1 (t), x2 (t) + εx∗2 (t), x3 (t) + εx∗3 (t)) − → → =− x (t) + εx∗ (t) 652 S. Özkaldı, K. İlarslan, Y. Yaylı − → → in D31 is differentiable. We call the real part − x (t) the indicatrix of x b (t). The dual arc → − length of the curve x b (t) from t1 to t is defined as sb := Zt Zt Z t D ′ E → ′ → → ′ → − − − b (t) dt = − x (t) dt + ε t , x∗ = s + εs∗ , x t1 t1 t1 − → − where t is a unit tangent vector of → x (t). From now on we will take the arc length s of → − x (t) as the parameter instead of t. → → − − → − The equalities relative to the derivatives of the dual Frenet vectors b t,n b, b b throughout the dual space curve are written in the matrix form as − → → − b t t 0 κ b 0 b d − → → − (2.1) = −ε ε κ b 0 τ b n b n b , 1 2 − db s − → → 0 −ε ε τ b 0 2 3 b b b b where κ b = κ + εκ∗ is the nowhere pure dual curvature, τb = τ + ετ ∗ is the nowhere pure dual torsion and D− D− D− D− → − →E →E D− →E → − → − → − →E → − →E →E D− → − b t, b t = ε1 , n b, n b = ε2 , b b, b b = ε3 , b t,n b = b t, b b = n b, b b = 0. The formulae (2.1) are called the Frenet formulae of the dual curve in dual Lorentzian n− n− →o n − →o → − → − →o → − space. The planes spanned by b t,b b , b t,n b and n b, b b at each point of the dual Lorentzian curve are called the rectifying plane, the osculating plane, and the normal plane, respectively [1, 2, 11, 14]. 3. Mannheim partner curves in D31 In this section, we define Mannheim partner curves in the dual space D31 and we give characterizations for Mannheim partner curves in the same space. 3.1. Definition. Let D31 be dual Lorentzian space with the Lorentzian inner product h·, ·i. If there exists a correspondence between the dual Lorentzian space curves α b and βb such that, at the corresponding points of the dual Lorentzian curves, the principal b then α normal lines of α b coincides with the binormal lines of β, b is called a dual Lorentzian b Mannheim curve, and βb a dual Lorentzian Mannheim partner curve of α b. The pair {b α, β} is said to be a dual Lorentzian Mannheim pair. Let α b : x b(b s) be a dual Lorentzian Mannheim curve in D31 parameterized by its arc length sb and βb : x b1 (b s1 ) the dual Lorentzian Mannheim partner curve with an arc length → − → − → − b parameter b s1 . Denote by t (b s), n b (b s), b b (b s) the dual Lorentzian Frenet frame field along α b:x b(b s). In the following theorems, we give necessary and sufficient conditions for a dual space curve to be a dual Lorentzian Mannheim curve. → − → − → − 3.2. Theorem. Let α b:x b(b s) be a Lorentzian curve in D31 and b t (b s), n b (b s) and b b (b s) the tangent, principal normal and binormal vector fields of α b:x b(b s), respectively. → − → − → − i) In the case when b t (b s) is a timelike vector, n b (b s) and b b (b s) are spacelike vectors, then α b:x b(b s) is a dual Lorentzian Mannheim curve if and only if its curvature κ b b τ2 − κ b is a never pure dual and torsion τb satisfy the formula κ b = λ(b b2 ), where λ constant. On Mannheim Partner Curves in Dual Lorentzian Space 653 → − → − → − ii) In the case when b t (b s) and n b (b s) are spacelike vectors, b b (b s) a timelike vector, then α b:x b(b s) is a dual Lorentzian Mannheim curve if and only if its curvature κ b b κ2 − τb2 ), where λ b is a never pure dual and torsion τb satisfy the formula κ b = λ(b constant. → − → − → − iii) In the case when b t (b s) and b b (b s) are spacelike vectors, n b (b s) a timelike vector, then α b:x b(b s) is a dual Lorentzian Mannheim curve if and only if its curvature b κ2 + τb2 ), where λ b is a never pure κ b and torsion τb satisfy the formula κ b = −λ(b dual constant. Proof. i) Let α b : x b(b s) be a dual Lorentzian Mannheim curve in D31 with arc length parameter b s, and βb : x b1 (b s1 ) the dual Lorentzian Mannheim partner curve with arc length parameter sb1 . Then by the definition we can assume that → b s )− (3.1) x b1 (b s) = x b(b s) + λ(b n b (b s) b s). By taking the derivative of (3.1) with respect to for some never pure dual constant λ(b s and applying the Frenet formulas we have b − → → dλ b− → b − db x1 (b s) bκ b = 1 + λb t + n b + λb τb b. db s db s → − → − Since b t1 is coincident with b b1 in direction, we get b s) dλ(b = 0. db s b is a never pure dual constant. Thus we have This means that λ − → − → db x1 (b s) bκ b bτ b = 1 + λb t + λb b. db s On the other hand, we have − → db − → − → db x1 db s s bκ b bτ b b t1 = = 1 + λb t + λb b . db s db s1 db s1 By taking the derivative of this equation with respect to b s1 and applying the Frenet formulas we obtain → − → → db → db t1 db s κ− τ− s b db bκ2 − λb bτ 2 − b db b b = λ t + κ b + λb n b +λ b db s db s1 db s db s db s1 2 − → db − → s bκ b bτ b + 1 + λb t + λb b . db s1 From this equation we get s bκ2 − λb bτ 2 db κ + λb b = 0, db s1 b τ2 − κ κ b = λ(b b2 ). This completes the proof. ii) Let α b:x b(b s) be a dual Lorentzian Mannheim curve in D31 with arc length parameter b s and β : x b b1 (b s1 ) the dual Lorentzian Mannheim partner curve with arc length parameter s1 . Then by the definition we can assume that b → b s )− (3.2) x b1 (b s) = x b(b s) + λ(b n b (b s) 654 S. Özkaldı, K. İlarslan, Y. Yaylı b s). By taking the derivative of (3.2) with respect to for some never pure dual constant λ(b s and applying the Frenet formulas we have b − → → dλ b− → b − db x1 (b s) bκ b = 1 − λb t + n b + λb τb b. db s db s → − → − Since b t1 is coincident with b b1 in direction, we get b s) dλ(b = 0. db s b is a never pure dual constant. Thus we have This means that λ − → − → db x1 (b s) bκ b bτ b = 1 − λb t + λb b. db s On the other hand, we have − → db − → − → db x1 db s s bκ b bτ b b t1 = = 1 − λb t + λb b . db s db s1 db s1 By taking the derivative of this equation with respect to b s1 and applying the Frenet formulas we obtain → − → → db → s db κ− db τ− s db t1 db 2 2 − b b b b b b = −λ t + κ b − λb κ + λb τ n b +λ b db s db s1 db s db s db s1 2 − → db − → s bκ b bτ b + 1 − λb t + λb b . db s1 From this equation we get s bκ2 + λb bτ 2 db κ − λb b = 0, db s1 b κ2 − τb2 ). κ b = λ(b This completes the proof. iii) Let α b:x b(b s) be a dual Lorentzian Mannheim curve in D31 with arc length parameter s, and βb : x b b1 (b s1 ) the dual Lorentzian Mannheim partner curve with arc length parameter s1 . Then by the definition we can assume that b → b s )− (3.3) x b1 (b s) = x b(b s) + λ(b n b (b s) b s). By taking the derivative of (3.3) with respect to for some never pure dual constant λ(b s and applying the Frenet formulas we have b − → → dλ b− → b − db x1 (b s) bκ b = 1 + λb t + n b + λb τb b. db s db s → − → − Since b t1 is coincident with b b1 in direction, we get b s) dλ(b = 0. db s b is a never pure dual constant. Thus we have This means that λ − → − → db x1 (b s) bκ b bτ b = 1 + λb t + λb b. db s On the other hand, we have − → db − → − → db x1 db s s bκ b bτ b b t1 = = 1 + λb t + λb b . db s db s1 db s1 On Mannheim Partner Curves in Dual Lorentzian Space 655 By taking the derivative of this equation with respect to b s1 and applying the Frenet formulas, we obtain → − → → db → db t1 db s κ− τ− s b db bκ2 + λb bτ 2 − b db b b = λ t + κ b + λb n b +λ b db s db s1 db s db s db s1 2 − → db − → s bκ b bτ b + 1 + λb t + λb b . db s1 From this equation we get s bκ2 + λb bτ 2 db = 0, κ + λb b db s1 b τ2 + κ κ b = −λ(b b2 ). This completes the proof. 3.3. Theorem. Let α b:x b(b s) be a dual Lorentzian Mannheim curve in D31 with arc length parameter sb. → − → − → − i) In the case where b t (b s) is a timelike vector, n b (b s) and b b (b s) are spacelike vectors, then βb : x b1 (b s1 ) is the dual Mannheim partner curve of α b if and only if the curvature κ b1 and the torsion τb1 of βb satisfy the following equation db τ1 κ b1 = 1−µ b2 τb12 db s1 µ b for some never pure dual constant µ b. → − → − → − b ii) In the case where t (b s) and n b (b s) are spacelike vectors, b b (b s) a timelike vector, then βb : x b1 (b s1 ) is the dual Mannheim partner curve of α b if and only if the curvature κ b1 and the torsion τb1 of βb satisfy the following equation db τ1 κ b1 =− 1+µ b2 τb12 db s1 µ b for some never pure dual constant µ b. → − → − → − b b iii) In the case where t (b s) and b (b s) are spacelike vectors, n b (b s) a timelike vector, then βb : x b1 (b s1 ) is the dual Mannheim partner curve of α b if and only if the curvature κ b1 and the torsion τb1 of βb satisfy the following equation db τ1 κ b1 =− 1−µ b2 τb12 db s1 µ b for some never pure dual constant µ b. Proof. i) Suppose that α b : x b(b s) is a dual Lorentzian Mannheim curve. Then by the definition we can assume that → − (3.4) x b(b s1 ) = x b1 (b s1 ) + µ b(b s 1 )b b1 (b s1 ) for some function µ b(b s1 ). By taking the derivative of (3.4) with respect to b s1 and applying the Frenet formulas, we have → → db − → − → − s db µ− b b (3.5) t = b t1+ b −µ bτb1 n b 1. db s1 db s1 → − → − Since b b is coincident with n b in direction, we get db µ = 0. db s1 656 S. Özkaldı, K. İlarslan, Y. Yaylı This means that µ b is a never pure dual constant. Thus we have → db − → − → − s b (3.6) t = b t 1−µ bτb1 n b 1. db s1 On the other hand, we have → − − → → − b b (3.7) t = b t 1 cosh θb + n b 1 sinh θ, → − → − where θb is the dual hyberbolic angle between b t and b t 1 at the corresponding points of b By taking the derivative of this equation with respect to sb1 , we obtain α b and β. ! ! → − → − → db − → − s dθb dθb κ bn b = b κ1 + sinh θb b t1+ κ b1 + cosh θb n b 1 + τb1 sinh θb b b 1. db s1 db s1 db s1 − → → − From this equation and the fact that the direction of n b is coincident with that of b b 1, we get ! dθb κ b1 + cosh θb = 0. db s1 Therefore we have dθb = −b κ1 . (3.8) db s1 − → → − From (3.6) and (3.7) and noting that b t 1 is orthogonal to b b 1 , we find that 1 µ bτb1 db s = =− . db s1 cosh θb sinh θb Then we have b µ bτb1 = − tanh θ. By taking the derivative of this equation and applying (3.8), we get db τ1 =κ b1 1 − µ b2 τb12 , µ b db s1 that is db τ1 κ b1 = 1−µ b2 τb12 . db s1 µ b Conversely, if the curvature κ b1 and torsion τb1 of the dual Lorentzian curve βb satisfy db τ1 κ b1 = 1−µ b2 τb12 db s1 µ b for some never pure dual constant µ b(b s), then we define a dual curve by → − (3.9) x b(b s1 ) = x b1 (b s1 ) + µ bb b1 (b s1 ) , and we will prove that α b is a dual Lorentzian Mannheim curve and that βb is the dual Lorentzian partner curve of α b. By taking the derivative of (3.9) with respect to sb1 twice, we get → db − → − → − s b (3.10) t = b t 1−µ bτb1 n b 1, db s1 2 2 → − → d sb − → − → db − → s db τ1 − (3.11) κ bn b + b t 2 = −b µκ b1 τb1 b t1+ κ b1 − µ b n b1−µ bτb12 b b 1, db s1 db s1 db s1 On Mannheim Partner Curves in Dual Lorentzian Space 657 respectively. Taking the cross product of (3.10) with (3.11) and noticing that we have (3.12) κ b1 − µ b db τ1 −µ b2 κ b1 τb12 = 0, db s1 3 → db − − → → − s = −b µ2 τb13 b t 1+µ bτb12 n b1 κ bb b db s1 By taking the cross product of (3.12) with (3.10), we obtain also 4 → − → db − s κ bn b = −b µτb12 1 − µ b2 τb12 b b 1. db s1 → − This means that the principal normal direction n b of α b:x b(b s) coincides with the binormal → − b b direction b 1 of β : x b1 (b s1 ). Hence α b : x b(b s) is a dual Lorentzian Mannheim curve and βb : x b1 (b s1 ) is its dual Lorentzian Mannheim partner curve. ii) Suppose that α b:x b(b s) is a dual Lorentzian Mannheim curve. Then by the definition we can assume that → − (3.13) x b(b s1 ) = x b1 (b s1 ) + µ b(b s 1 )b b1 (b s1 ) for some function µ b(b s1 ). By taking the derivative of (3.13) with respect to sb1 and applying the Frenet formulas, we have → → db − → − → − db µ− s b b = b t1+ b +µ bτb1 n b 1. (3.14) t db s1 db s1 → − → − Since b b is coincident with n b in direction, we get db µ = 0. db s1 This means that µ b is a never pure dual constant. Thus we have → db − → − → − s b (3.15) t = b t 1+µ bτb1 n b 1. db s1 On the other hand, we have → − − → → − b b (3.16) t = b t 1 cos θb + n b 1 sin θ, → − → − b where θb is the dual angle between b t and b t 1 at the corresponding points of α b and β. By taking the derivative of this equation with respect to sb1 , we obtain ! ! → − → − → db − → − s dθb dθb b b κ bn b =− κ b1 + sin θ t 1 + κ b1 + cos θb n b 1 + τb1 sin θb b b 1. db s1 db s1 db s1 − → → − From this equation and the fact that the direction of n b is coincident with that of b b 1, we get b dθ κ b1 + db sin θb = 0 s1 b d θ κ b1 + dbs1 cos θb = 0. Therefore we have dθb (3.17) = −b κ1 . db s1 658 S. Özkaldı, K. İlarslan, Y. Yaylı − → → − From (3.15), (3.16) and noticing that b t 1 is orthogonal to b b 1 , we find that db s 1 µ bτb1 = = . b db s1 cos θ sin θb Then we have b µ bτb1 = tan θ. By taking the derivative of this equation and applying (3.17), we get db τ1 µ b = −b κ1 1 + µ b2 τb12 , db s1 that is db τ1 κ b1 =− 1+µ b2 τb12 . db s1 µ b Conversely, if the curvature κ b1 and torsion τb1 of the dual Lorentzian curve βb satisfy db τ1 κ b1 =− 1+µ b2 τb12 db s1 µ b for some never pure dual constant µ b(b s), then we define a dual curve by → − (3.18) x b(b s1 ) = x b1 (b s1 ) + µ bb b1 (b s1 ) , and we will prove that α b is a dual Lorentzian Mannheim curve and that βb is the dual Lorentzian partner curve of α b. By taking the derivative of (3.18) with respect to sb1 twice, we get → − → db − → − s b (3.19) = b t 1+µ bτb1 n b 1, t db s1 2 2 → − → d sb − → − → → db − s db τ1 − (3.20) κ bn b + b t 2 = −b µκ b1 τb1 b t1+ κ b1 + µ b n b1+µ bτb12 b b 1, db s1 db s1 db s1 respectively. Taking the cross product of (3.19) with (3.20) and noticing that we have (3.21) −b κ1 − µ b db τ1 −µ b2 κ b1 τb12 = 0, db s1 3 → db − − → → − s κ bb b = −b µ2 τb13 b t 1+µ bτb12 n b 1. db s1 By taking the cross product of (3.21) with (3.19), we obtain also 4 → − → db − s κ bn b =µ bτb12 1 + µ b2 τb12 b b 1. db s1 → − This means that the principal normal direction n b of α b:x b(b s) coincides with the binormal → − direction b b 1 of βb : x b1 (b s1 ). Hence, α b :x b(b s) is a dual Lorentzian Mannheim curve and βb : x b1 (b s1 ) is its dual Lorentzian Mannheim partner curve. iii) Suppose that α b:x b(b s) is a dual Lorentzian Mannheim curve. Then by the definition we can assume that − → (3.22) x b(b s1 ) = x b1 (b s1 ) + µ b(b s 1 )b b1 (b s1 ) for some function µ b(b s1 ). By taking the derivative of (3.22) with respect to sb1 and applying the Frenet formulas, we have → → db − → − → − s db µ− b b (3.23) t = b t1+ b +µ bτb1 n b 1. db s1 db s1 On Mannheim Partner Curves in Dual Lorentzian Space 659 → − − → Since b b is coincident with n b in direction, we get db µ = 0. db s1 This means that µ b is a never pure dual constant. Thus we have → db − → − → − s b (3.24) t = b t 1+µ bτb1 n b 1. db s1 On the other hand, we have → − − → → − b b (3.25) t = b t 1 cosh θb + n b 1 sinh θ, where α b and → − → − θb is the dual hyberbolic angle between b t and b t 1 at the corresponding points of b By taking the derivative of this equation with respect to sb1 , we obtain β. ! ! → − → − → db − → − s dθb dθb b b κ bn b = b κ1 + sinh θ t 1 + κ b1 + cosh θb n b 1 + τb1 sinh θb b b 1. db s1 db s1 db s1 − → → − From this equation and the fact that the direction of n b is coincident with that of b b 1, we get ! dθb κ b1 + cosh θb = 0. db s1 Therefore we have dθb (3.26) = −b κ1 . db s1 − → → − From (3.24), (3.25) and noticing that b t 1 is orthogonal to b b 1 , we find that db s 1 µ bτb1 = = . db s1 cosh θb sinh θb Then we have b µ bτb1 = tanh θ. By taking the derivative of this equation and applying (3.26), we get that is µ b db τ1 = −b κ1 1 − µ b2 τb12 , db s1 db τ1 κ b1 =− 1−µ b2 τb12 . db s1 µ b Conversely, if the curvature κ b1 and torsion τb1 of the dual curve βb satisfy db τ1 κ b1 =− 1−µ b2 τb12 db s1 µ b for some never pure dual constant µ b(b s), then we define a dual Lorentzian curve by → − (3.27) x b(b s1 ) = x b1 (b s1 ) + µ bb b1 (b s1 ) , and we will prove that α b is a dual Lorentzian Mannheim curve and that βb is the dual Lorentzian partner curve of α b. 660 S. Özkaldı, K. İlarslan, Y. Yaylı By taking the derivative of (3.27) with respect to sb1 twice, we get → db − → − → − s b (3.28) t = b t 1+µ bτb1 n b 1, db s1 2 2 → − → d sb − → − → db − → s db τ1 − (3.29) κ bn b + b t 2 =µ bκ b1 τb1 b t1+ κ b1 + µ b n b1 +µ bτb12 b b 1, db s1 db s1 db s1 respectively. Taking the cross product of (3.28) with (3.29) and noticing that we have (3.30) κ b1 + µ b db τ1 −µ b2 κ b1 τb12 = 0, db s1 3 − db → → − → − s b κ bb =µ b2 τb13 b t 1 +µ bτb12 n b1 db s1 By taking the cross product of (3.30) with (3.28), we obtain also 4 → − → db − s κ bn b = −b µτb12 1 − µ b2 τb12 b b 1. db s1 → − This means that the principal normal direction n b of α b:x b(b s) coincides with the binormal → − direction b b 1 of βb : x b1 (b s1 ). Hence, α b :x b(b s) is a dual Lorentzian Mannheim curve and βb : x b1 (b s1 ) is its dual Lorentzian Mannheim partner curve. 3.4. Remark. Let α b = α + εα∗ and βb = β + εβ ∗ be non-null dual curves in the dual 3 Lorentzian space D1 . As a special case, if we consider the dual part of the given curves as α∗ = 0 and β ∗ = 0, then α b and βb are non-null Lorentzian curves in R31 . In this case all the results of this paper are similar to the results given in [9]. Acknowledgement We wish to thank the referee for the careful reading of the manuscript and for the constructive comments that have substantially improved the presentation of the paper. References [1] Ayyildiz, N., Çöken, A. C. and Yücesan, A. A characterization of dual Lorentzian spherical curves in the dual Lorentzian space, Taiwanese J. Math. 11 (4), 999–1018, 2007. [2] Ayyildiz, N., Çöken, A. C. and Yücesan, A. On the dual Darboux rotation axis of the spacelike dual space curve. Demonstratio Math. 37 (1), 197–202, 2004. [3] Çöken, A. C. and Görgülü, A. On the dual Darboux rotation axis of the dual space curve, Demonstratio Math. 35 (2), 385–390, 2002. [4] do Carmo, M. P. Differential Geometry of Curves and Surfaces (Prentice-Hall, Inc., Engelwood Cliffs, N. J., 1976). [5] Guggenheimer, W. Differential Geometry (McGraw-Hill, New York, 1963). [6] Hacısalihoğlu, H. H. On the pitch of a closed ruled surface, Mech. Mach. Theory 7 (3), 291–305, 1972. [7] Köse, Ö., Nizamoğlu, Ş. and Sezer, M. An explicit characterization of dual spherical curves, Doğa Mat. 12 (3), 105–113, 1988. [8] Kuhnel, W. Differential Geometry: Curves-Surfaces-Manifolds (Braunschweig, Wiesbaden, 1999). [9] Liu, H. and Wang, F. Mannheim partner curves in 3-space, J. Geom. 88, 120–126, 2008. [10] O’Neill, B. Semi-Riemannian Geometry with Applications to Relativity (Academic Press, London, 1983). [11] Özbey, E. and Oral, M. A Study on rectifying curves in the dual Lorentzian space, Bull. Korean Math. Soc. 46 (5), 967–978, 2009. On Mannheim Partner Curves in Dual Lorentzian Space 661 [12] Özkaldı, S., İlarslan, K. and Yaylı, Y. On Mannheim partner curves in dual space, An. Şt. Univ. Ovidius Constanta 17 (2), 131–142, 2009. [13] Struik, D. J. Differential Geometry, Second Ed. (Addison-Wesley, Reading, Massachusetts, 1961). [14] Turgut, M. On the invariants of time-like dual curves, Hacet. J. Math. Stat. 37 (2), 129–133, 2008. [15] Uğurlu H. H., and Çalışkan, A. The Study mapping for directed spacelike and timelike lines in Minkowski 3-space R31 , Mathematical and Computational Applications 1 (2), 142–148, 1996. [16] Veldkamp, G. R. On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics, Mech. Mach. Theory 11 (2), 141–156, 1976. [17] Yücesan, A., Ayyıldız, N. and Çöken, A. C. On rectifying dual space curves, Rev. Mat. Complut. 20 (2), 497–506, 2007. [18] Yücesan, A., Çöken, A. C. and Ayyildiz, N. On the dual Darboux rotation axis of the timelike dual space curve, Balkan J. Geom. Appl. 7 (2), 137–142, 2002.
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Department of Mathematics, Faculty of Science, Ankara University, Tandoğan, Ankara,
Turkey. E-mail:
(S. Özkaldı) [email protected] (Y. Yaylı) [email protected]