8_459_alagoz.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 4, 2009, (564-573) ISSN 1307-5543 – www.ejpam.com On the Transformations Preserving Asymptotic Directions of Hypersurfaces in the Euclidean Space Yasemin Alagöz∗ and Ziya Soyuçok Department of Mathematics, Faculty of Science and Letters, Yildiz Technical University, Istanbul, Turkey Abstract. We consider the transformations preserving asymptotic directions of hypersurfaces in n-dimensional Euclidean space and we obtain a system of equations which must be satisfied by transformations. 2000 Mathematics Subject Classifications: 53A05. Key Words and Phrases: Hypersurfaces, asymptotic directions. 1. Introduction In the Euclidean space, the projective transformation preserves the asymptotic lines of a surface [3]. In [4] the inverse of that problem is considered and it is obtained that the most transformation preserving the asymptotic lines of surfaces in 3-dimensional Euclidean space is the projective one. But that paper has very long ∗Corresponding author. Email addresses: ygulluk�yildiz.edu.tr (Y. Alagöz), zsoyu ok�yildiz.edu.tr (Z. Soyuçok) http://www.ejpam.com 564 c© 2009 EJPAM All rights reserved. Y. Alagöz and Z. Soyuçok / Eur. J. Pure Appl. Math, 2 (2009), (564-573) 565 calculations and it seems very difficult to generalize for the n-dimensional space by using given method. Moreover, since it has some errors that transformation is not the general projective transformation [1]. In this paper, we consider the transformations which preserve the asymptotic di- rections of hypersurfaces in n-dimensional Euclidean space and we obtain a system of equations. The transformations must satisfy these equations system. 2. The Equation of the Asymptotic Directions of a Hypersurface In the n-dimensional Euclidean space, a hypersurface can be expressed by the equation r � u1, . . . , un−1 � = (x1 � u1, . . . , un−1 � , x2 � u1, . . . , un−1 � , . . . , x n � u1, . . . , un−1 � ) (1) where the metric of the space is given by ds2 = (d x1)2 + (d x2)2+ . . .+ (d x n)2. (2) We assume that r � u1, u2, . . . , un−1 � is a differentiable function of order 3 and the tan- gent vectors r,1, r,2, . . . , r,n−1 of the hypersurface are linearly independent where r,i ≡ ∂ r ∂ ui , (i = 1, 2, . . . , n− 1) . (3) The first and second fundamental forms of the hypersurface are I = gi jduidu j, I I = Li jduidu j, � i, j = 1, 2, . . . , n− 1 � (4) where gi j = r,i.r, j (5) Li j = r,i j.N, � r,i j ≡ ∂ 2r ∂ ui∂ u j � . (6) Y. Alagöz and Z. Soyuçok / Eur. J. Pure Appl. Math, 2 (2009), (564-573) 566 Here N is the unit normal vector of the hypersurface, that is, r,i.N= 0 (7) and N.N= 1. (8) The differential equation of the asymptotic directions of the hypersurface is given by Li jduidu j = 0 (9) [2, p.44 ] and [5, p.134 ]. The system (7) can be written as ANT = 0 (10) where A=         x1 ,1 x2 ,1 · · · x n ,1 x1 ,2 x2 ,2 · · · x n ,2 ... ... · · · ... x1 ,n−1 x2 ,n−1 · · · x n ,n−1         , � x k ,i = ∂ x k ∂ ui � (11) and N= � N1, N2, . . . , Nn � . (12) Since the vectors r,i = � x1 ,i , x2 ,i . . . , x n ,i � , (i = 1, 2, . . . , n− 1) (13) are linearly independent, we can assume that ∆n = det         x1 ,1 x2 ,1 · · · x n−1 ,1 x1 ,2 x2 ,2 · · · x n−1 ,2 ... ... · · · ... x1 ,n−1 x2 ,n−1 · · · x n−1 ,n−1         6= 0. (14) Y. Alagöz and Z. Soyuçok / Eur. J. Pure Appl. Math, 2 (2009), (564-573) 567 Then from (10) and (8) we have N= 1 k � ∆1,−∆2, . . . , (−1)1+n∆n � (15) where ∆i is the determinant of the matrix which is obtained by omitting ith column in the coefficients matrix A and k = p ∆2 1+∆ 2 2+ . . .+∆2 n . (16) Accordingly, from (6) we get Li j = 1 k [x1 ,i j ∆1− x2 ,i j ∆2+ . . .+ (−1)1+n x n ,i j ∆n] (17) and so kLi j = det         x1 ,i j x1 ,1 x1 ,2 · · · x1 ,n−1 x2 ,i j x2 ,1 x2 ,2 · · · x2 ,n−1 ... ... ... · · · ... x n ,i j x n ,1 x n ,2 · · · x n ,n−1         , � x k ,i j = ∂ 2x k ∂ ui∂ u j � . (18) Now for a hypersurface S let us choose the parameters as u1 = x1, u2 = x2, . . . , un−1 = x n−1. (19) Then, the equation of S becomes r(x1, x2, ..., x n−1) = (x1, x2, ..., x n−1, x n(x1, ..., x n−1)) (20) and from (18) we get kLi j = (−1) n+1 x n ,i j . (21) See also [2, p.36 ]. The differential equation of the asymptotic directions of S, from (9), is obtained as x n ,i j d x id x j = 0 � i, j = 1, 2, . . . , n− 1 � . (22) Y. Alagöz and Z. Soyuçok / Eur. J. Pure Appl. Math, 2 (2009), (564-573) 568 3. Conditions for a Transformation Preserving the Asymptotic Directions Here we determine transformations preserving the asymptotic directions of a hy- persurface. In the n-dimensional Euclidean space let us consider the coordinate trans- formation T : ya = ya � x1, x2, . . . , x n � , (a = 1, 2, . . . , n) . (23) We assume that T is differentiable of order 3 and ∆= det h T,1 T,2 · · · T,n i = � � � T,1 T,2 · · · T,n � � � 6= 0 (24) where T,b =         y1 ,b y2 ,b ... yn ,b         , � ya ,b = ∂ ya ∂ x b ; b = 1, 2, . . . , n � . (25) If the transformation T is applied to the hypersurface S which is defined by the equa- tion (20), then we get T ′ : ya = ya � x1, x2, . . . , x n−1, x n � x1, x2, . . . , x n−1 �� . (26) So the transformation T transforms the hypersurface S to a hypersurface S∗ which is given by the equation r∗ � x1, x2, . . . , x n−1 � = � y1, y2, . . . , yn � (27) where ya = ya(x1, x2, . . . , x n−1, x n � x1, x2, . . . , x n−1 � ), (a = 1, . . . , n) . For the hypersurface S∗, k∗L∗ i j = � � � T ′ ,1 T′ ,2 · · · T′ ,n−1 T′ ,i j � � � (28) Y. Alagöz and Z. Soyuçok / Eur. J. Pure Appl. Math, 2 (2009), (564-573) 569 is obtained from (18), where T ′ ,i = T,i + T,nx n ,i , T ′ ,i j = T,i j + T,inx n , j + T,nj x n ,i + T,nnx n ,i x n , j + T,nx n ,i j , (29) and T,i j =         y1 ,i j y2 ,i j ... yn ,i j         , � ya ,i j = ∂ 2 ya ∂ x i∂ x j ; i, j = 1, 2, . . . , n− 1 � . (30) Using (29) and (30), from (28) we can write k∗L∗ ii = � � � T,1T,2 ... T,n−1T,ii � � �+ � � � T,nT,2 ... T,n−1T,ii � � � x n ,1 + � � � T,1T,n ... T,n−1T,ii � � � x n ,2 + ...+ � � � T,1T,2 ... T,nT,n−1T,ii � � � x n ,n−2 + � � � T,1T,2 ... T,n−2T,nT,ii � � � x n ,n−1 + 2 � � � T,1T,2 ... T,n−1T,in � � � x n ,i +2 � � � T,nT,2 ... T,n−1T,in � � � x n ,i x n ,1 + 2 � � � T,1T,n ... T,n−1T,in � � � x n ,i x n ,2 +...+ 2 � � � T,1T,2 ... T,nT,n−1T,in � � � x n ,i x n ,n−2 +2 � � � T,1T,2 ... T,n−2T,nT,in � � � x n ,i x n ,n−1 + � � � T,1T,2 ... T,n−1T,nn � � � (x n ,i )2 + � � � T,nT,2 ... T,n−1T,nn � � � (x n ,i )2 x n ,1 + � � �T,1T,n ... T,n−1T,nn � � � (x n ,i )2 x n ,2 +...+ � � �T,1T,2 ... T,nT,n−1T,nn � � � (x n ,i )2 x n ,n−2 + � � � T,1T,2 ... T,n−2T,nT,nn � � � (x n ,i )2 x n ,n−1 +∆.x n ,ii (31) and k∗L∗ i j = � � � T,1T,2 ... T,n−1T,i j � � �+ � � � T,nT,2 ... T,n−1T,i j � � � x n ,1 + � � � T,1T,n ... T,n−1T,i j � � � x n ,2 + ...+ � � � T,1T,2 ... T,nT,n−1T,i j � � � x n ,n−2 + � � � T,1T,2 ... T,n−2T,nT,i j � � � x n ,n−1 + � � � T,1T,2 ... T,n−1T,nj � � � x n ,i + � � � T,nT,2 ... T,n−1T,nj � � � x n ,i x n ,1 + � � � T,1T,n ... T,n−1T,nj � � � x n ,i x n ,2 Y. Alagöz and Z. Soyuçok / Eur. J. Pure Appl. Math, 2 (2009), (564-573) 570 +...+ � � � T,1T,2 ... T,nT,n−1T,nj � � � x n ,i x n ,n−2 + � � � T,1T,2 ... T,n−2T,nT,nj � � � x n ,i x n ,n−1 + � � � T,1T,2 ... T,n−1T,in � � � x n , j + � � � T,nT,2 ... T,n−1T,in � � � x n , j x n ,1 + � � � T,1T,n ... T,n−1T,in � � � x n , j x n ,2 +...+ � � � T,1T,2 ... T,nT,n−1T,in � � � x n , j x n ,n−2 + � � � T,1T,2 ... T,n−2T,nT,in � � � x n , j x n ,n−1 + � � � T,1T,2 ... T,n−1T,nn � � � x n ,i x n , j + � � � T,nT,2 ... T,n−1T,nn � � � x n ,i x n , j x n ,1 + � � � T,1T,n ... T,n−1T,nn � � � x n ,i x n , j x n ,2 +...+ � � �T,1T,2 ... T,nT,n−1T,nn � � � x n ,i x n , j x n ,n−2 + � � �T,1T,2 ... T,n−2T,nT,nn � � � x n ,i x n , j x n ,n−1 +∆.x n ,i j . (32) The differential equation of the asymptotic directions of S∗, according to (9), is L∗ i j d x id x j = 0. (33) In order that the transformation T transforms the asymptotic directions of the hyper- surface S to the asymptotic directions of the hypersurface S∗ it must transform the equation (22) to the equation (33). Accordingly, our conditions are L∗ i j = t x n ,i j (34) where t is an arbitrary function of the variables x1, x2, . . . , x n−1 . The equations (31) and (32) can be written as follows: k∗L∗ ii = ∆0(ii) +∆1(ii)x n ,1 +∆2(ii)x n ,2 + ...+∆n−1(ii)x n ,n−1 +2[∆0(in) +∆1(in)x n ,1 +∆2(in)x n ,2 + ...+∆n−1(in)x n ,n−1 ]x n ,i +[∆0(nn) +∆1(nn)x n ,1 +∆2(nn)x n ,2 + ...+∆n−1(nn)x n ,n−1 ](x n ,i )2 +∆.x n ,ii (35) and k∗L∗ i j = ∆0(i j) +∆1(i j)x n ,1 +∆2(i j)x n ,2 + ...+∆n−1(i j)x n ,n−1 Y. Alagöz and Z. Soyuçok / Eur. J. Pure Appl. Math, 2 (2009), (564-573) 571 +[∆0(in) +∆1(in)x n ,1 +∆2(in)x n ,2 + ...+∆n−1(in)x n ,n−1 ]x n , j +[∆0( jn) +∆1( jn)x n ,1 +∆2( jn)x n ,2 + ...+∆n−1( jn)x n ,n−1 ]x n ,i +[∆0(nn) +∆1(nn)x n ,1 +∆2(nn)x n ,2 + ...+∆n−1(nn)x n ,n−1 ]x n ,i x n , j +∆.x n ,i j (36) where∆0(ab) denotes the determinant which is obtained by replacing the nth column with T,ab in the determinant ∆ which is defined by (24), and ∆k(ab) denotes the determinant which is obtained by replacing the nth column with T,ab and kth column with T,n in the determinant ∆. For example, ∆2(44) = � �T1TnT3 . . .Tn−1T44 � � . The equations (34) must be satisfied by any hypersurface. So, using the quantities given by (35) in (34) we have the following conditions: ∆1(nn) = 0,∆2(nn) = 0, . . . ,∆n−1(nn) = 0, (37) ∆0(ii) = 0,∆1(ii) = 0, . . . ,∆i−1(ii) = 0,∆i+1(ii) = 0, . . . ,∆n−1(ii) = 0, (38) ∆1(in) = 0,∆2(in) = 0, . . . ,∆i−1(in) = 0,∆i+1(in) = 0, . . . ,∆n−1(in) = 0, (39) ∆i(ii) + 2∆0(in) = 0, ∆0(nn) + 2∆i(in) = 0. (40) From (37) and (38) we respectively get T,nn = 2AnT,n (41) and T,ii = 2AiT,i (42) and so T,bb = 2AbT,b (43) Y. Alagöz and Z. Soyuçok / Eur. J. Pure Appl. Math, 2 (2009), (564-573) 572 where A1, A2, . . . , An are arbitrary functions of variables x1, x2, . . . , x n. Using (43) in (39) and (40), we have T,in = AnT,i + AiT,n. (44) Now let us use the quantities given by (36) in (34) which must be satisfied by any hypersurface. Then we have the following conditions: ∆0(i j) = 0,∆1(i j) = 0, ...,∆i−1(i j) = 0,∆i+1(i j) = 0, (45) ...,∆ j−1(i j) = 0,∆ j+1(i j) = 0, ...,∆n−1(i j) = 0 ∆i(i j) +∆0( jn) = 0, ∆ j(i j) +∆0(in) = 0, (46) ∆i(in) +∆ j( jn) +∆0(nn) = 0 (47) and (37) and (39) again. From (44) and (45), using (46) we get T,i j = A jT,i + AiT, j. (48) The results (43), (44) and (48) can be expressed by a single equation as T,ab = AbT,a + AaT,b, (a, b = 1, 2, . . . , n) . (49) (47) is automatically satisfied by these results. From (37) to (40) and from (45) to (47) all equations are satisfied by (49). Thus we have the following theorem. Theorem 1. A transformation T which preserves the asymptotic directions of a hyper- surface must satisfy the equations T,ab = AbT,a + AaT,b, (a, b = 1, 2, . . . , n) (50) where A1, A2, . . . , An are arbitrary functions of variables x1, x2, . . . , x n . REFERENCES 573 References [1] Y. Alagöz, Z. Soyuçok, A note “On transformation preserving asymptotic lines of surfaces in Three-Dimensional Euclidean Space”. Bulletin of the Technical University of Istanbul. Submitted. [2] Y. Aminov, The Geometry of Submanifolds. Gordon and Breach Science Publisher, Ams- terdam, 2001, p.36-45. [3] L. P Eisenhart, A Treatise on The Differantial Geometry of Curves and Surfaces. Dover Publications, Inc., New York, 1960, p.202. [4] F. Uras, On transformation preserving asymptotic lines of surfaces in Three-Dimensional Euclidean Space. Bulletin of the Technical University of Istanbul. 48 , 1: 165-179 (1995). [5] C. E. Weatherburn, An Introduction to Riemannian Geometry and The Tensor Calculus. Cambridge University Press, Cambridge, 1963, p.134.