ON THE PARALLEL SURFACES IN GALILEAN SPACE

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1 ON THE PARALLEL SURFACES IN GALILEAN SPACE Mustafa Dede 1, Cumali Ekici 2 and A. Ceylan Çöken 3 1 Kilis 7 Aral k University, Department of Mathematics, 79000, Kilis-TURKEY 2 Eskişehir Osmangazi University, Department of Mathematics and Computer Science, 26480, Eskişehir-TURKEY 3 Süleyman Demirel University, Department of Mathematics, 32260, Isparta-TURKEY August 11, 2012 Abstract In this paper, rst of all, the de nition of parallel surfaces in Galilean space is given. Then, the relationship between the curvatures of the parallel surfaces in Galilean space is determined. Moreover, the rst and second fundamental forms of parallel surfaces are found in Galilean space. Consequently, we obtained Gauss curvature and mean curvature of parallel surface in terms of those curvatures of the base surface. Key words: Parallel surfaces, Curvatures of surface, Galilean space. 1 Introduction It is known that two surfaces with a common normal are called parallel surfaces. A large number of papers and books have been published in the literature which deal with parallel surfaces in both Minkowski space and Euclidean space such as [1,6,10-11,14]. However, this paper presents the di erential properties of the parallel surfaces in three-dimensional Galilean space. There are nine related plane geometries including Euclidean geometry, hyperbolic geometry and elliptic geometry. Galilean geometry is one of these geometries whose motions are the Galilean transformations of classical kinematics [13]. Di erential geometry of the Galilean space G 3 and especially the geometry of ruled surfaces in this space have been largely developed in O. Röschel s paper [12]. Some more results about ruled surfaces in G 3 have been given in [7-9]. A. Ö¼grenmiş [15] obtained characterizations of helix for a curve with respect to the Frenet frame in Galilean space. Curves in pseudo-galilean space have 1

2 been explained in details in [2]. Recently, C. Cekici and M. Dede [4] investigated Darboux vectors of ruled surface in pseudo-galilean space. The Galilean space G 3 is a Cayley Klein space equipped with the projective metric of signature (0; 0; +; +). The absolute gure of the Galilean geometry consists of an ordered triple f!; f; Ig, where! is the real (absolute) plane, f is the real line (absolute line) in!. I is the xed elliptic involution of points of f. De nition 1.1. A plane is called Euclidean if it contains f, otherwise it is called isotropic. Planes x = constant are Euclidean and so is the plane!. Other planes are isotropic. A vector u = (u 1 ; u 2 ; u 3 ) is said to be non-isotropic if u 1 6= 0. All unit non-isotropic vectors are of the form u = (1; u 2 ; u 3 ). For isotropic vectors, u 1 = 0 holds [8]. Since x = 0 plane is Euclidean in Galilean space, it is easy to see that isotropic vectors are on the Euclidean plane. De nition 1.2. Let a = (x; y; z) and b = (x 1 ; y 1 ; z 1 ) be vectors in Galilean space. The scalar product is de ned by < a; b >= x 1 x (1) The norm of a is de ned by kak = jxj ; and a is called a unit vector if kak = 1: On the other hand, as a consequence of De nition 1.1, we de ne the scalar product of two isotropic vectors, p = (0; y; z) and q = (0; y 1 ; z 1 ); as < p; q > 1 = yy 1 + zz 1 (2) The orthogonality of isotropic vectors, p? 1 q; means that < p; q > 1 = 0: The norm of p is de ned by kpk 1 = p y 2 + z 2 ; and p is called a unit isotropic vector if kpk 1 = 1 [13]: De nition 1.3. Let u = (u 1 ; u 2 ; u 3 ) and v = (v 1 ; v 2 ; v 3 ) be vectors in Galilean space [8]. The cross product of the vectors u and v is de ned as follows: u ^ v = 0 e 2 e 3 u 1 u 2 u 3 v 1 v 2 v 3 = (0; u 3v 1 u 1 v 3 ; u 1 v 2 u 2 v 1 ) (3) De nition 1.4. Let " be a plane and f(") the intersection of the absolute line f and ": Figure 1 2

3 The point f(") is called the absolute point of ". Then f(")? = I(f(")) denotes the point on f orthogonal to f(") according to the elliptic involution I. This is an elliptic involution because there is no line perpendicular to itself [12]. The elliptic involution in homogeneous coordinates is given by (0 : 0 : x 2 : x 3 )! (0 : 0 : x 3 : x 2 ) (4) De nition 1.5. If an admisable curve C of the class C r (r 3) is given by the parametrization r(x) = (x; y(x); z(x)) then x is a Galilean invariant of the arc length on C [7]. associated invariant moving trihedron is given by t = (1; y 0 (x); z 0 (x)); In Figure 2, the n = 1 (0; y00 (x); z 00 (x)); (5) b = 1 (0; z00 (x); y 00 (x)) where = p y 00 (x) 2 + z 00 (x) 2 is the curvature and = 1 2 det[r0 (x); r 00 (x); r 000 (x)] is the torsion. Figure 2 Frenet formulas may be written as 2 d 4 t 3 2 n 5 = dx b t n b 2 Surface Theory in Galilean Space 3 5 Assume that M is a surface in G 3. parametrization The equation of M is given by the '(v 1 ; v 2 ) = (x(v 1 ; v 2 ); y(v 1 ; v 2 ); z(v 1 ; v 2 )); v 1 ; v 2 2 R (6) 3

4 where x(v 1 ; v 2 ); y(v 1 ; v 2 ); z(v 1 ; v 2 ) 2 C 3. The isotropic unit normal vector eld N, shown in Figure 3, is given by N = ' ;1 ^ ' ;2 (0; z ;2 x ;1 z ;1 x ;2 ; y ;1 x ;2 y ;2 x ;1 ) ';1 ^ ' 1 = p (7) ;2 (z;1 x ;2 z ;2 x ;1 ) 2 + (y ;2 x ;1 y ;1 x ;2 ) 2 where partial di erentiation with respect to v 1 and v 2 will be denoted by su xes 1 and 2 respectively, that ' ;1 ; v 2 1 and ' ;2 ; v 2 2 [12]. Figure 3 Using (4); we obtain the isotropic unit vector in the tangent plane of surface as = (0; y ;1x ;2 y ;2 x ;1 ; z ;1 x ;2 z ;2 x ;1 ) (8) w where hn; i 1 = 0; 2 = 1; w = ';1 ^ ' 1 ;2 (9) by means of Galilean geometry. Observe that a straightforward computation shows that can be expressed by = x ;2' ;1 x ;1 ' ;2 w (10) where x ;1 and x ;2 are the partial di erentiation of the rst component of the surface M with respect to v 1 and v 2, respectively. x ;1 ; v 2 1 ; x ;2 ; v 2 2 (11) Consequently to simplify the presentation (10), we may use Einstein summation convention, then may rewritten as follows = g i ' ;i = g 1 ' ;1 + g 2 ' ;2 (12) 4

5 where g 1 = x ;1 g 2 = x ;2 g ij = g i g j g 1 = x ;2 w g2 = x (13) ;1 w gij = g i g j From De nition 1.2, the rst fundamental form I of the surface is given by I = (g ij + h ij )dv i dv j (14) where h ij and g ij (i; j = 1; 2) are called induced metric on the surface given by h ij = ' ;i ; ' ; g ;j 1 ij = ' ;i ; ' ;j (15) and 8 < 0; dv 1 : dv 2 non-isotropic = : 1; dv 1 : dv 2 isotropic (16) Isotropic curves, shown in Figure 4, are the intersections of the surface M with Euclidean planes [12]. All other curves on the surface are called nonisotropic curves. Figure 4 Let (s) = '(v(s) 1 ; v(s) 2 ) be a non-isotropic curve in a surface patch '; parametrized by the arc length s: From (13) it follows that g i v i0 = 1 (17) where " 0 " refers to d ds : The coe cients L ij of second fundamental form are given by ';ij x ;1 x ;ij ' ;1 L ij = ; N x ;1 1 (18) 5

6 and the Christo el symbols of the surface are given by ';ij x 1 ;2 x ;ij ' ;2 ';ij x 2 ;1 x ;ij ' ;1 ij = ; ij = w w Theorem 2.1. Let M be a surface in Galilean space. ; 1 ; 1 (19) ' ;ij = k ij' ;k + L ij N (20) is called the Gauss equation of the surface [12]. Theorem 2.2. Let M be a surface in Galilean space. The Weingarten equation is given by N ;i = B i + C i N (21) where C i = 0; B i = g k L ki [12]. Moreover, from (9) and (21), we have ;i = g k L ki N (22) Theorem 2.3. Let (s) = '(v(s) 1 ; v(s) 2 ) be non-isotropic curve on the surface, parametrized by the arc length s: The equation of normal curvature k n and geodesic curvature k g of the surface are given by, respectively k n = L ij v i0 v j0 ; k g = k ij vi0 v j0 + v k00 g k (23) where " 0 " refers to d ds. In addition, let be the Euclidean angle between the isotropic vectors, the surface normal N and the curve normal n; we have cos = hn; '00 i 1 k' 00 k 1, sin = h; '00 i 1 k' 00 k 1 (24) Consequently, k n and k g are obtained by, respectively k n = cos, k g = sin (25) where = k' 00 k 1 is the curvature of (s) [12]. Corollary 2.4. The equation of the asymptotic lines are given by L ij v i0 v j0 = 0 (26) Corollary 2.5. Since K 1 corresponding value of the normal curvature may be found by making use of Lagrange s multipliers, we have This implies the following theorem. K 1 = L 11L 22 (L 12 ) 2 w 2 g ij L ij (27) 6

7 Theorem 2.6. Let M be a surface in G 3 [12]. The Gauss curvature K and the mean curvature H of the surface are given by, respectively K = det L ij w 2 ; 2H = g ij L ij (28) The following corollary is clear from (27) and (28). Corollary 2.7. K 1 can be expressed by K 1 = K 2H (29) 3 Parallel Surfaces in Galilean Space De nition 3.1. Let M and M be two surfaces in Galilean space G 3 and 2 R; 8p 2 M: The function f : '(v 1 ; v 2 )! ' (v 1 ; v 2 ) p! f(p) = [p 1 ; p 2 + a 2 (p); p 3 + a 3 (p)] is called the parallelization function between M and M where p = (p 1 ; p 2 ; p 3 ) and N = a i = (0; a 2 ; a 3 i i=2 is the isotropic unit normal vector eld on M and furthermore M is called parallel surface to M in G 3 where is a given positive real number. Figure 5 In Figure 5, T and T are the isotropic tangent planes of parallel surfaces M and M, respectively. 7

8 Note that from the de nition of parallel surfaces, we have N(p) = N (f(p)). Moreover this leads to the fact that (p) = (f(p)) in Galilean space: De nition 3.2. Let M and M be parallel surfaces in Galilean space. We de ne the parallel surface M to base surface M at distance as ' (v 1 ; v 2 ) = '(v 1 ; v 2 ) + N (30) where N is normal vector of the base surface. Theorem 3.3. Let M and M be parallel surfaces in Galilean space. The relationship between the w = ' ;1 ^ ' 1 ;2 and w = ' ;1 ^ ' ;2 1 can be given as follows: w = w(1 2H) (31) Proof: Taking the partial derivatives of M gives ' ;1 = ' ;1 + N ;1, ' ;2 = ' ;2 + N ;2 (32) Thus, by (3), we see N ;1 ^ N ;2 = 0. In addition, by (21), (28) and (32), we get ' ;1 ^ ' ;2 = (' ;1 ^ ' ;2 )(1 2H) Taking norm of the both sides, we have w = w(1 2H) Theorem 3.4. Let M and M be parallel surfaces in Galilean space. The rst fundamental form I of the parallel surface is given by 8 < I dv 1 : dv 2 non-isotropic I = : I (2L ij g k L ki g k L kj )dv i dv j dv 1 : dv 2 isotropic Proof: Let us now consider = 0 in (14), it follows that I = g ijdv i dv j From (7), (11) and (30), we obtained the partial di erentiation of the rst component of the surface M as which implies that By using (13), I is obtained by x ;1 = x ;1 ; x ;2 = x ;2 (33) g i = g i (34) I = g ij dv i dv j = I (35) We now consider = 1: In this case, the rst fundamental form I is I = h ijdv i dv j (36) 8

9 Di erentiating (30), then using (15) gives ' ;i ; ';j = h ij + 2 N ;i ; ' ;j hn ;i ; N ;j i 1 (37) 1 Finally, substituting, (12), (14), (20) and (21) into (37), we have I = I (2L ij g k L ki g k L kj )dv i dv j Theorem 3.5. Let M and M be two parallel surfaces. The coe cients L ij of second fundamental form of the parallel surface are given by Proof: Di erentiating (30), we get L ij = L ij g k L ki g k L kj (38) ' ;ij = ' ;ij + N ;ij (39) Substituting (33) and (39) into (18) and using hn ;ij ; Ni 1 = hn ;i ; N ;j i 1 implies that L ij = L ij hn ;i ; N ;j i 1 From (9) and (21), L ij is obtained by L ij = L ij g k L ki g k L kj (40) Corollary 3.6. Asymptotic lines of the parallel surface M are given by L ij = L ij g k L ki g k L kj = 0 (41) Theorem 3.7. Let M and M be two parallel surfaces in G 3 and (s) = ' (v(s) 1 ; v(s) 2 ) be a non-isotropic curve on the parallel surface; parametrized by the arc length s; given by g i v i0 = 1 (42) where " 0 " refers to d ds. The normal curvature k n of parallel surface is given by k n = k n (g k L ki g k L kj )v i0 v j0 (43) where k n is the normal curvature of M: Proof: Di erentiating (30) with respect to s gives and ' 0 = ' ;i v i0 + N ;i v i0 (44) ' 00 = ' ;ij v i0 v j0 + ' ;k v k00 + (N ;ij v i0 v j0 + N ;k v k00 ) (45) Substituting (20) into (45), we get ' 00 = ( k ijv i0 v j0 + v k00 )' ;k + L ij v i0 v j0 N + (N ;ij v i0 v j0 + N ;k v k00 ) (46) 9

10 Taking scalar product of both sides of (46) with N gives ' 00 ; N 1 = (L ij + hn; N ;ij i 1 )v i0 v j0 Using (21), (23) and hn ;ij ; Ni 1 = hn ;i ; N ;j i 1 implies that the relation between the normal curvatures of two parallel surfaces is k n = k n (g k L ki g k L kj )v i0 v j0 (47) Theorem 3.8. Let M and M be two parallel surfaces in G 3. The relation between the geodesic curvatures of two parallel surfaces is given by k g = k g g k L ki v k00 (48) Proof: Taking scalar product of both sides of (46) with gives ' 00 ; 1 = ( k ijv i0 v j0 + v k00 ) ' ;k ; 1 + (hn ;ij; i 1 v i0 v j0 + hn ;k ; i 1 v k00 ) (49) Substituting (21) and (23) into (49), we have k g = k g + (hn ;ij ; i 1 v i0 v j0 g k L ki v k00 ) (50) From (21) and (22) we have hn ;ij ; i 1 = 0 which implies that k g = k g g k L ki v k00 Theorem 3.9. Let M and M be two parallel surfaces in G 3 : The relations between the Gauss curvatures and the mean curvatures of two parallel surfaces are K K = (51) 1 2H and H = H 1 2H respectively. Proof: Substituting (31) and (38) into (28) gives Simple calculation implies that (52) K = det[l ij g k L ki g k L kj ] w 2 (1 2H) 2 (53) K = (L 11L 22 L 2 12)(1 (g 11 L g 12 L 12 + g 22 L 22 )) w 2 (1 2H) 2 (54) Combining (28) and (54), we have K = K 1 2H (55) 10

11 Using (13), (31) and (33) implies that (g ij ) = Taking account of (28), (38) and (56), we nd that 2H = g ij (1 2H) 2 (56) g ij (1 2H) 2 (L ij g k L ki g k L kj ) (57) Finally, substituting (28) into (57) then, H can be written as H = H 1 2H (58) Now we shall consider some particular cases of the results (55) and (58). Theorem Let M and M be two parallel surfaces in G 3 : If base surface is minimal, then parallel surface is minimal. Proof: Since M is minimal surface, H = 0: Therefore, from (58) we have H = 0 Theorem Let M and M be two parallel surfaces in G 3 : If base surface is Weingarten, then parallel surface is Weingarten. Proof: Since base surface is Weingarten, it satis es the following condition H ;1 K ;2 H ;2 K ;1 = 0 (59) On the other hand, di erentiating (55) and (58) with respect to v 1 and v 2 ; we get K ;1 = (1 2H)K ;1 + 2KH ;1 (1 2H) 2 ; H ;1 = (1 2H)H ;1 + 2HH ;1 (1 2H) 2 (60) K ;2 = (1 2H)K ;2 + 2KH ;2 (1 2H) 2 ; H ;2 = (1 2H)H ;2 + 2HH ;2 (1 2H) 2 (61) Thus, Combining (59) and (62) gives This completes the proof. H ;1K ;2 H ;2K ;1 = H ;1K ;2 H ;2 K ;1 (1 2H) 3 (62) H ;1K ;2 H ;2K ;1 = 0 11

12 References: [1] Çöken, A. C., Çiftçi, Ü. and Ekici, C. On parallel timelike ruled surfaces with timelike rulings, Kuwait Journal of Science & Engineering, 35(1A), 21-31, [2] Divjak, B. Curves in pseudo-galilean geometry, Annales Univ. Sci. Budapest, 41, , [3] Eisenhart, L. P. A treatise on the di erential geometry of curves and surfaces (Dover Publication, Inc., New York, 1960). [4] Ekici, C. and Dede, M. On the Darboux vector of ruled surfaces in pseudo- Galilean space, Mathematical and Computational Applications, 16(4), , [5] Guggenheimer, H. W. Di erential geometry (McGraw-Hill, New York, 1963). [6] Sa¼gel, M. K., and Hac saliho¼glu, H. H. On the parallel hypersurfaces with constant curvature, Commun. Fac. Sci. Univ. Ank. Series A, 40, [7] Kamenarovic, I. Existence theorems for ruled surfaces in the Galilean space G 3 ; Rad Hrvatske Akad. Znan. Umj. Mat., 456(10), , [8] Milin-Sipus, Z. Ruled Weingarten surfaces in Galilean space, Periodica Mathematica Hungarica, 56(2), , [9] Milin-Sipus, Z. Divjak, B. Special curves on ruled surfaces in Galilean and pseudo-galilean spaces, Acta Math. Hungar., 98(3), , [10] Roberts, S. On Parallel surfaces, Proc. London Math. Soc., 1-4(1), , [11] Roberts, S. On the parallel surfaces of developables and curves of double curvature, Proc. London Math. Soc., 1-5(1), 90-94, [12] Röschel, O. Die geometrie des Galileischen raumes (Habilitationsschrift, Leoben, 1984). [13] Yaglom, I. M. A simple non-euclidean geometry and its physical basis (Springer-Verlag, New York Inc., 1979). [14] Görgülü, A. and Çöken A. C. The Euler theorem for parallel pseudo- Euclidean hpersurfaces in pseudo-euclidean space E1 n+1, Journ. Inst. Math. and Comp. Sci. (Math. Series), 6.2, , [15 ] Ö¼grenmiş, A., Ergüt, M. and Bektaş, M. On the helices in the Galilean space G 3, Iranian Journal of Science & Technology, Transaction A, 31(A2), ,