Erratum to: "Ruled surfaces with timelike rulings" [Appl. Math. Comput. 147: (2004) 241-253]

In this paper, using [E. Kasap, _ I. Aydemir, N. Kuruoğlu, Erratum to: ‘‘Ruled surfaces with timelike rulings’’ [Appl. Math. Comput. 147 (2004) 241–253], Applied Mathematics and Computation 168 (2005) 1461–1468 [2]], some mistakes which are related to the classification maximal ruled surfaces with timelike rulings in the last section of [Applied Mathematics and Computation 147 (2004) 241–253 [1]] have been found and corrected then re-written. 2005 Published by Elsevier Inc. 2.4. Maximal timelike ruled surfaces A classical result of Catalan states that the only ruled minimal surface in Euclidean 3-space E are the plane and the helicoid. As a continuation of the results obtained by Woestijne [3] on maximal timelike ruled surface, we introduced a new technique different of that used in his study to give a classification of maximal timelike ruled surfaces. The mean curvature H of the surfaces in Lorentz space is given by H 1⁄4 e 2 habg where g ab are the components of the contravarient metric tensor. It is easy to see that, the mean curvature of the ruled surfaces (I) is given by 0096-3 doi:10 DO E-m H 1⁄4 1 2 2k ~L0;~L0 ‘1 þ ~a0 ^~Lþ v~L0 ^ L; j~e2 þ v~L00 1 2v ~a0;~L0 v2 ~L0;~L0 ‘21 3=2 2 4 3 5. ð2:12Þ In [3], we have proved that for a maximal timelike ruled surface, the generator must be coincident with the principal normal field of its base curve. Then, the ruled surface (I) may be a maximal if ~LðsÞ 1⁄4~e2 i.e. ‘1 = ‘3 = 0 and ‘2 = 1. In this case, one can show that, the ruled surface (I) is a timelike ruled surface. From (1.2), (2.1) and (2.12), one can see that the timelike ruled surface (I) is maximal if and only if 003/$ see front matter 2005 Published by Elsevier Inc. .1016/j.amc.2005.11.082 I of original article: 10.1016/S0096-3003(02)00664-1 ail address: [email protected] Erratum / Applied Mathematics and Computation 179 (2006) 402–405 403 H 1⁄4 1 2 v s0 þ v s0j sj0 ð Þ 1 2vj v2ðj2 þ s2Þ ð Þ 1⁄4 0 or equivalently s 0 = 0 and js 0 sj 0 = 0, "v 2 IR. This is valid if j and s are constants. From (2.2), the striction curve of the maximal timelike ruled surface with timelike rulings is given by ~bðsÞ 1⁄4~aðsÞ þ ðj=ðj þ sÞÞ~e2ðsÞ. From (1.2) it follows that ~b 1⁄4~bðsÞ is a spacelike curve if jjj > jsj or jsj > jjj. From these inequalities it is easy to see that then, the parametrization of maximal timelike ruled surfaces with timelike rulings is given as Class I: jjj = jsj 5 0 uðs; vÞ 1⁄4 j j2 þ s2 þ v sh ffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2 þ s2 p s; j j2 þ s2 þ v ch ffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2 þ s2 p s; s ffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2 þ s2 p s . Class II: jjj = jsj = 0Since~LðsÞ 1⁄4~e2, and from (1.2) one can see that~L00ðsÞ 1⁄4 0 i.e.,~aıvðsÞ 1⁄4 0. That is,~aðsÞ is a polynomial of degree 3. Then, we have uðs; vÞ 1⁄4 X3 i1⁄41 ai s i þ v X3 i1⁄42 iði 1Þai s 2 ! . Thus, we have the proof of the following theorem: Theorem 7. Every maximal timelike ruled surface with timelike rulings, whose ruling is not null, is a congruent to a part of one of the following surfaces: (i) Time-like plane. (ii) Helicoid of third kind (jjj = jsj 5 0). (iii) Conjugate of Enneper’s surface of second kind (jjj = jsj = 0). The types of maximal timelike ruled surfaces (ii) and (iii) of this theorem are ploted as examples which are shown in Figs. 1 and 2. Fig. 1. Helicoid of third kind. Fig. 2. Conjugate of Enneper’s surface of second kind. Fig. 3. Timelike ruled surface. 404 Erratum / Applied Mathematics and Computation 179 (2006) 402–405 Example. For the ruled surface uðs; vÞ 1⁄4 ffiffiffi 2 p cos s 2 ffiffiffi 2 p v sin s; sþ 3v; ffiffiffi 2 p sin sþ 2 ffiffiffi 2 p v cos s it is easy to see that aðsÞ 1⁄4 ð ffiffiffi 2 p cos s; s; ffiffiffi 2 p sin sÞ and~LðsÞ 1⁄4 ð 2 ffiffiffi 2 p sin s; 3; 2 ffiffiffi 2 p cos sÞ are the base curve (spacelike) and the generator (timelike). The striction curve is the base curve. This surface is a timelike ruled surface for which k = 1/2 (Fig. 3).