Stability of closed hyperruled surfaces

Some geometers have been interested in differential geometry of the variational problems connected with general surfaces. During the last few decades, this interest increased rapidly as more researchers became involved and gained results. Specifically one may cite, in this regard, the works of B.Y. Chen [J. London Math. Soc. 6 (2) (1973) 321; Total Mean Curvature and Submanifolds of Finit Type, World Scientific, Sigapore, 1984; Geometry of Submanifolds, Marcel Dekker, New York, 1973], M.A. Soliman et al. [Bull. Fac. Sci., Assiut Univ., 24 (2-c) (1995) 189], T.J. Willmore [Total Curvature in Riemannian Geometry, Ellis Horwood, Chichester, UK, 1982; Topics in Differential Geometry, Academic Press, New York, 1976, p. 149] and J. Weiner [Math. J. 27 (1978) 19]. In this paper which consists of five sections, the variational differential geometric methods to the analysis of hyperruled surfaces in En+1 is a completely new issue and is not encountered in any of the relevant literature. In the second and third sections, we investigate the general case when the base curve of the hyperruled surface is not an orthogonal trajectory of the generating space and the fourth section is devoted to the investigation of the special case when the base curve of the hyperruled surface is an orthogonal trajectory of the generating space. Finally, as an application for the stability of compact ruled surfaces, several examples immersed in E3 and E4 are considered and the effect of the normal variations on the ruled surfaces is translated to figures.