The tangential variation on hyperruled surfaces

Abstract

One of the most interesting and profound aspects of classical differential geometry is its interplay with the calculus of variations. In fact, the main differential geometric ideas of the calculus of variation occur over and over again and are continually being invented and rediscovered in a vast array of classical and modern differential geometry. So the normal variational problem on general surfaces and hyperruled surfaces were studied by some geometers, specifically one may cite, in this regard, the works of Abdel-All and Abd-Ellah [Chaos, Solitons & Fractals, in press; Ind. J. Pure Appl. Math., in press; Adv. Model. Anal. 37 (2000)], Abdel-All and Hussien [J. Korean Math. Soc. 3 (1999) 663], Chen [J. Lond. Math. Soc. 6 (1973) 321; Total Mean Curvature and Submanifolds of Finite Type, World Scientific Publishing copre Ltd, 1984; Geometry of Submanifolds, Marcel Dekker, New York, 1973], Soliman et al. [Bull. Fac. Sci., Assiut Univ. 24 (1995) 189], Willmore [Total Curvature in Riemannian Geometry, Ellis Horwod Limited Publishers, 1982]. The purpose of the present work is to study effectiveness the tangential variation in the direction of the tangent for the base curve of hyperruled surfaces in E n+1 .