International Journal of Differential Equations and Applications —————————————————————————— Volume 7 No. 3 2003, 339-348 ON THE MINIMALITY AND TOTAL DEVELOPABILITY OF THE TIME-LIKE RULED SURFACES WITH THE TIME-LIKE GENERATING SPACE IN THE MINKOWSKI SPACE IR n 1 S. Can 1 § , B. Uyar 2 , ˙ I. Aydemir 3 1,2,3 Department of Mathematics Science and Arts Faculty Ondokuz Mayıs University Kurupelit 55139, Samsun, TURKEY 3 e-mails: iaydemir@omu.edu.tr Abstract: The purpose of this paper is, first, to introduce a summary of known results and the definition of the time-like ruled surface with the time-like generating space in the Minkowski space IR n 1 (Section 1); second, to present some characteristic results related with minimality and total developability of the ruled surface in the Minkowski space IR n 1 (Section 2). AMS Subject Classification: 53A10, 53A35 Key Words: Minkowski space, time-like ruled surface, minimality, total developability 1. Introduction We will assume throughout that this paper that all manifolds, maps, vector fields, etc. are differentiable of class C ∞ . First of all, we give some properties of a general submanifold M of the Minkowski n−space IR n 1 , [2]. Let D be a Levi-Civita connection of IR n 1 and D be a Levi-Civita connection of M . If X, Y ∈ χ(M ) and V is the second fundamental tensor of M , we have by decomposing D X Y Received: August 11, 2003 c  2003 Academic Publications § Correspondence author