Existence Theorems for Timelike Ruled Surfaces in Minkowski -Space Mehmet Önder Celal Bayar University, Faculty of Science and Arts, Department of Mathematics, Muradiye Campus, 45047 Muradiye, Manisa, Turkey. E-mail: mehmet.onder@bayar.edu.tr Abstract In this study we investigate the existence theorems for timelike ruled surfaces in Minkowski -space. We obtained a general system and give the existence theorems for timelike ruled surface according to Gaussian curvature, distribution parameter and strictional distance. Moreover, we give some special cases such as the directirx of the surface is a geodesic, an asymptotic line, a line of curvature or a general helix. MSC: 5B5, 5C50. Key words: Minkowski -space; timelike ruled surface; Frenet frame.. Introduction In the space, a continuously moving of a straight line generates a surface which is called ruled surface. Ruled surfaces have the most important positions and applications in the study of design problems in spatial mechanisms and physics, kinematics and computer aided design (CAD). So, these surfaces are one of the most important topics of surface theory. Because of this position of ruled surfaces, geometers have studied on these surfaces in Euclidean space and they have investigated many properties of the ruled surfaces [4,6,0]. Moreover, Minkowski space IR is more interesting than the Euclidean space. In this space, curves and surfaces have different casual Lorentzian characters such as timelike, spacelike or null (lightlike). For example, a continuously moving of a timelike line along a curve generates a timelike ruled surface. Turgut and Hacısalihoğlu have studied timelike ruled surfaces in Minkowski -space and given some properties of these surfaces []. Timelike ruled surface with timelike rulings have been studied by Abdel-All and others []. Küçük has obtained some results on the developable timelike ruled surfaces in the same space [8]. Furthermore, Uğurlu and Önder have introduced Frenet frames and Frenet invariants of timelike ruled surfaces in IR []. In this study we give existence theorems for timelike ruled surfaces in Minkowski -space by using a similar procedure given in [5]. We obtained a general system and give the existence of developable timelike ruled surfaces. Moreover, we give some special cases such as the directirx of the surface is a geodesic, an asymptotic line, a line of curvature or a general helix.. Preliminaries The Minkowski -space IR is the real vector space Lorentzian flat metric given by, = dx + dx + dx IR IR provided with standard where ( x, x, x) is a standard rectangular coordinate system of IR. An arbitrary vector v = ( v, v, v) in IR can have one of three Lorentzian causal characters; it can be spacelike if v, v > 0 or v = 0, timelike if v, v < 0 and null (lightlike) if v, v = 0 and v 0. Similarly, an arbitrary curve α = α( s) can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors α ( s) are spacelike, timelike or null (lightlike), respectively [9]. We
say that a timelike vector is future pointing or past pointing if the first compound of vector is positive or negative, respectively. The norm of the vector v = ( v, v, v) is given by v = v, v. For any vectors x = ( x, x, x) and y = ( y, y, y) in IR, Lorentzian vector product of x and y is defined by e e e x y = x x x = ( x y x y, x y x y, x y x y ). y y y Definition..(See [,]) i) Hyperbolic angle: Let x and y be future pointing (or past pointing) timelike vectors in IR. Then there is a unique real number θ 0 such that < x, y >= x y coshθ. This number is called the hyperbolic angle between the vectors x and y. ii) Central angle: Let x and y be spacelike vectors in IR that span a timelike vector subspace. Then there is a unique real number θ 0 such that < x, y >= x y coshθ. This number is called the central angle between the vectors x and y. iii) Spacelike angle: Let x and y be spacelike vectors in IR that span a spacelike vector subspace. Then there is a unique real number θ 0 such that < x, y >= x y cosθ. This number is called the spacelike angle between the vectors x and y. iv) Lorentzian timelike angle: Let x be a spacelike vector and y be a timelike vector in IR. Then there is a unique real number θ 0 such that < x, y >= x y sinhθ. This number is called the Lorentzian timelike angle between the vectors x and y.. Timelike Ruled Surfaces in Minkowski -space Let I be an open interval in the real line IR, k = k ( s) be a curve in IR defined on I and q = q( s) be a unit direction vector of an oriented timelike line in IR. Then we have following parametrization for a timelike ruled surface N r ( s, v) = k ( s) + v q( s). () The parametric s -curve of this surface is a straight timelike line of the surface which is called ruling. For v = 0, the parametric v -curve of this surface is k = k ( s) which is called base curve or generating curve of the surface. In particular, if the direction of q is constant, then the ruled surface is said to be cylindrical, and non-cylindrical otherwise. The distribution parameter (or drall) of N is given by k, q, q d = () q, q dk dq where k =, q = (see [,]). If k, q, q = 0, then normal vectors are collinear at all ds ds points of same ruling and at nonsingular points of the surface N, the tangent planes are identical. We then say that tangent plane contacts the surface along a ruling. Such a ruling is
called a torsal ruling. If k, q, q 0, then the tangent planes of the surface N are distinct at all points of same ruling which is called nontorsal []. Definition.. A timelike ruled surface whose all rulings are torsal is called a developable timelike ruled surface. The remaining timelike ruled surfaces are called skew timelike ruled surfaces. Then, from () it is clear that a timelike ruled surface is developable if and only if at all its points the distribution parameter d = 0 []. For the unit normal vector m of a timelike ruled surface we have rs rv ( k + vq ) q m = = r s r v k, q q, q k + vq, k + vq Then, at the points of a nontorsal ruling s = s we have q q a = lim m( s, v) = v q The plane of a skew timelike ruled surface N which passes through its ruling s and is perpendicular to the vector a is called asymptotic plane α. The tangent plane γ passing through the ruling s which is perpendicular to the asymptotic plane α is called central plane. The point C of the ruling s where asymptotic plane is perpendicular to central plane is called central point of the ruling s. The set of central points of all rulings is called striction curve of the surface. The parametrization of the striction curve c = c( s) on a timelike ruled surface is given by q, k c( s) = k ( s) + v0q( s) = k q q, q q, k where v0 = is called strictional distance (For details []). q, q Theorem..(Chasles Theorem): Let the base curve of a timelike ruled surface be its striction curve. For the angle µ between tangent plane of timelike ruled surface at the point ( s, v 0) of a nontorsal ruling s and central plane we have tan µ = v0 / d where d is the distribution parameter of ruling s, v 0 is strictional distance and central point has the coordinates ( s,0) []. 4. Existence Theorems for Timelike Ruled Surfaces Let N be timelike ruled surface in IR given by the parametrization r ( s, v) = k ( s) + v q( s) () where k = k ( s) is directrix of N, s is arc length of k ( s) and q( s) is a unit timelike vector field on N. The directrix k ( s) can be a timelike or spacelike curve. Let assume that k ( s) be a timelike curve. Then the Frenet formulae of k ( s) are given as follows
T 0 k 0 T N k 0 k = N B 0 k 0 B where k ( s) = T ( s), T, T =, N, N = B, B =, T, N = T, B = N, B = 0 (4), T, N, B are unit tangent, principal normal and binormal vectors, respectively, and k and k are curvature and torsion of timelike curve k ( s), respectively []. The unit normal vector m of the surface is a spacelike vector and can be given in the form m = cosϕ N + sinϕb (5) where ϕ = ϕ( s) is spacelike angle between spacelike unit vectors m and N. Let A be a spacelike unit vector in the tangent plane of the surface and be perpendicular to unit tangent T. Then we can represent the timelike ruling q in the form q = coshθt + sinhθ A (6) where θ = θ ( s) is hyperbolic angle between timelike unit vectors q and T. It is clear that the vector A lies on the plane Sp{ N, B}. Then we can write A = sinϕn + cosϕb (7) By differentiating (5) and (7) with respect to s it follows m = k cos ϕt + ( ϕ + k) A, A = k sin ϕt ( ϕ + k) m (8) respectively. Similarly, considering (7) and (8) the differentiation of (6) is obtained as follows q = sinh θ ( θ k sin ϕ) T + ( cosh θ ( k θ sin ϕ) ( ϕ + k) sinhθ cosϕ ) N (9) + θ coshθ cos ϕ ( ϕ + k ) sinhθ sinϕ B ( ) and (9) gives us ( q ) = q, q = θ k θ sin ϕ + k (cosh θ cos ϕ + sin ϕ) k ( + k )sinh cosh cos + ( + k ) sinh After these computations we can give the following theorems. ϕ θ θ ϕ ϕ θ Theorem 4.. For a timelike curve k ( s) as the directrix there exists a two-parameter family of timelike ruled surfaces with a given distribution parameter and a given strictional distance. Proof: The strictional distance and distribution parameter of a timelike ruled surface is given by k, q k, q, q v0 =, d = () q, q q, q respectively []. Then from (6) and (9) it follows sinh θ ( θ k sinh sin ϕ ) θ ( k coshθ cos ϕ ( ϕ + k) sinhθ ) v0 =, d = () ( q ) ( q ) From () we have (0) 4
d v θ k θ sin ϕ + k (cosh θ cos ϕ + sin ϕ) + = 0 ( θ k sin ϕ) From (0) and () it follows k ( + k )sinh cosh cos + ( + k ) sinh + ϕ θ θ ϕ ϕ θ ( θ k sin ϕ) d = θ k ϕ + v0 ( q ) ( sin ) (4) Substituting (4) in () we have v0 sinhθ θ = + k sinϕ d + v0 (5) d ϕ k k cothθ cosϕ = + d + v0 which are the determining equations for timelike ruled surfaces with a timelike directrix k ( s) and a given distribution parameter and a given strictional distance. That proves the theorem. Corollary 4.. For any timelike directrix k ( s) as the strictional line there exists a twoparameter family of timelike ruled surfaces with a given distribution parameter. Proof: If the directrix k ( s) θ = k sinϕ ϕ = k + k cothθ cosϕ d that finishes the proof. is strictional line then v 0 = 0. Thus (5) becomes Theorem 4.. For any timelike directrix k ( s) there exists a two-parameter family of timelike ruled surfaces with a given Gaussian curvature and a given angle between the tangent planes and central planes along k ( s). Proof: From Chasles theorem we have v0 tan µ = (7) d where µ is the spacelike angle between tangent plane and central plane of timelike ruled surface at the point ( s, v 0). Furthermore, Gaussian curvature K of a timelike ruled surface is given by d K = (8) ( d + v0 ) (See [7]). If we put d + v n = = 0 (9) K d then K defines n uniquely and followings holds d = nsin µ, v0 = nsin µ cos µ (0) By using (0), system (5) can be given in the form () (6) 5
θ = sinh θ cot µ + k sin ϕ n ϕ = k + k cothθ cosϕ n that finishes the proof. Theorem 4.4. For any timelike directrix k ( s) there exists a two-parameter family of general developable timelike ruled surfaces with a given strictional distance. Proof: If the timelike ruled surface N is developable and not a cylinder we have v0 0, d = 0. Then from (5) it follows sinhθ θ = + k sinϕ v0 () ϕ = k + k cothθ cosϕ which shows that there exists a two-parameter family of general developable timelike ruled surfaces with a given strictional distance. Theorem 4.5. For any timelike directrix k ( s) there exists a two-parameter family of timelike cylinders. Proof: If the timelike ruled surface N is a cylinder then the direction of the ruling q is constant and from (9) we have sinh θ ( θ k sin ϕ) = 0, cosh θ ( k θ sin ϕ) ( ϕ + k)sinhθ cosϕ = 0, θ coshθ cos ϕ ( ϕ + k) sinhθ sinϕ = 0, that gives us θ = k sin ϕ, ϕ = k + k cothθ cosϕ () and () shows that there exists a two-parameter family of timelike cylinders. 5. Some Special Cases In this section we consider some special cases such as the directirx k ( s) is a geodesic, an asymptotic line or a line of curvature. Then we can give the followings. Theorem 5.. For any timelike directrix k ( s) there exists exactly one timelike ruled surface with a given Gaussian curvature on which k ( s) is a geodesic. Proof: Let the directrix k ( s) be a geodesic. Then we have N = ± m. By considering (5) we have ϕ = aπ, ( a Z ). From system () it follows nk tanhθ = nk + which shows that if the directrix k ( s) is a geodesic then there exists exactly one timelike ruled surface with a given Gaussian curvature. () 6
Theorem 5.. For any timelike directrix k ( s) there exists a one-parameter family of timelike ruled surfaces with a given angle between the tangent planes and corresponding central planes on which k ( s) is an asymptotic line. Proof: Assume that the directrix k ( s) is an asymptotic line. Then from (5) we have ϕ = π / and from the system () we have θ = sinhθ cot µ + k, n = n k which determines a one-parameter family of timelike ruled surfaces. Theorem 5.. Let the angles θ and µ be constants. Then a timelike directrix curve k ( s) is an asymptotic line on a timelike ruled surface N if and only if k ( s) is a general helix. Proof: Let the directrix k ( s) be an asymptotic line on N. Then ϕ = π /. Since the angles θ and µ are constants from () it follows k = sinhθ cot µ k which is constant i.e. k ( s) is a general helix. Conversely, if the angles θ and µ are constants and k ( s) is a general helix then from () we have ϕ = π / which shows that k ( s) is an asymptotic line on N. Theorem 5.. A timelike curve k ( s) is a line of curvature on a timelike ruled surface N if and only if ϕ ( s) = k( s) ds + C (4) holds where C is a constant. Proof: A curve k ( s) is a line of curvature if and only if surface normals along k ( s) generate a developable ruled surface, i.e. iff it is k, m, m = 0 (5) Then from (8) we have T, m, ( ϕ + k) A = 0 which gives ϕ = k and (4) is obtained. Theorem 5.4. For any timelike curve k ( s) there exists a one-parameter family of timelike ruled surfaces with a given Gaussian curvature along k ( s) on which k ( s) is a line of curvature. Proof: Since k ( s) is a line of curvature (4) holds and from () we have nk = tanhθ secϕ (6) that finishes the proof. Theorem 5.5. If θ is constant and timelike directrix k ( s) is a line of curvature on a timelike ruled surface N then there exists the following relationship between the angles ϕ and µ 7
tanϕ = coshθ cot µ (7) Proof: Since θ is constant and k ( s) is a line of curvature, from () and (4) we have k sin ϕ = sinh cot, k cos tanh n θ µ ϕ = n θ (8) which gives (7). 5. Conclusions The existence theorems of timelike ruled surfaces are given by considering a timelike directrix. Of course, one can obtain corresponding theorems for a timelike ruled surface with a spacelike directrix or for a spacelike ruled surface. References [] Abdel-All, N.H., Abdel-Baky, R.A., Hamdoon, F.M.: Ruled surfaces with timelike rulings. App. Math. and Comp. 47, 4 5 (004). [] Aydoğmuş, Ö., Kula, L., Yaylı, Y.: On point-line displacement in Minkowski -space. Differ. Geom. Dyn. Syst. 0, -4 (008). [] Birman, G., Nomizo, K.: Trigonometry in Lorentzian Geometry. Ann. Math. Mont. 9, (9), 54-549 (984). [4] Guggenheimer, H.: Differential Geometry. McGraw-Hill Book Comp. Inc. London, Lib. Cong. Cat. Card No. 68-8, (96). [5] Kamenarovic, I.: Existence Theorems for Ruled Surfaces in the Galilean Space G. Rad Hrvatske Akad. Znan. Umj. Mat. 0 (456), 8-86 (99). [6] Karger, A., Novak, J.: Space Kinematics and Lie Groups. STNL Publishers of Technical Lit., Prague, Czechoslovakia (978). [7] Kazaz, M., Özdemir, A., Güroğlu, T.: On the determination of a developable timelike ruled surface. SDÜ Fen-Edebiyat Fakültesi Fen Dergisi (E-Dergi), (), 7-79, (008). [8] Küçük, A.: On the developable timelike trajectory ruled surfaces in Lorentz -space IR. App. Math. and Comp. 57, 48-489 (004). [9] O Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London (98). [0] Ravani, B., Ku, T.S.: Bertrand Offsets of ruled and developable surfaces. Comp. Aided Geom. Design. (), 45-5 (99). [] Turgut, A., Hacısalihoğlu, H.H.: Timelike ruled surfaces in the Minkowski -space. Far East J. Math. Sci. 5 (), 8 90 (997). [] Uğurlu, H.H., Önder, M.: Instantaneous Rotation vectors of Skew Timelike Ruled Surfaces in Minkowski -space. VI. Geometry Symposium, Uludağ University, Bursa, Turkey. (008). [] Walrave, J.: Curves and surfaces in Minkowski space. PhD. thesis, K. U. Leuven, Fac. of Science, Leuven (995). 8