Geometry of constant slope curves and submanifolds

Abstract

The thesis entitled Geometry of constant slope curves and submanifolds newlineincludes seven chapters. Each chapter of the thesis is summarized as follows: newlineThe first chapter is an introduction that contains history, motivation, basic newlineideas, and outcomes related to Bertrand partner curves, WC newlineand#8727; partner curves, spherical newlineindicatrix of a curve, Smarandache ruled surfaces, canal surfaces, rotational embedded newlinesurfaces, and invariant surfaces in Sol3 and N il3 spaces based on an extensive overview newlineof the literature. newlineChapter II entitled Bertrand partner D-curves and spherical indicatrix newlinein E newline3 newlineis devoted to the study of the new parametrization of Bertrand partner D-curves newlineand spherical indicatrix of a new approach to Bertrand curve in Euclidean 3-space. newlineThe Frenet approach of moving frames to study curves in Euclidean spaces led newlineDarboux to introduce the Darboux frame to study surfaces. This frame is a natural newlinemoving frame to characterize curves on surfaces. In 2016, Kazaz et al. [68] defined newlinethe Bertrand pair curve {and#945;, and#945;and#8727;} lying on the surfaces and called these new associated newlinecurves as Bertrand partner D-curves by using the Darboux frame of the curves. In newline2020, Camci et al. [32] introduced a new relationship between a pair of Bertrand newlinecurves and#945; and and#945; newlineand#8727; as follows: newlineand#945; newlineand#8727; newline(s newlineand#8727; newline) = and#945;(s) + u1(s) T(s) + v1(s) N(s) + w1(s) B(s), (0.0.1) newlinewhere u1(s), v1(s) and w1(s) are differentiable functions and (T, N, B) the Frenet newlineframe of and#945;. After studying the concept of a new relationship between Bertrand pair newlinecurves satisfying (0.0.1) in Euclidean 3-space [32], it is natural to investigate Bertrand ... newline