The Invariants of Dual Parallel Equidistant Ruled Surfaces

Abstract

In this paper, we calculate the Gaussian curvatures of the dual spherical indicatrix curves formed on unit dual sphere by the Blaschke vectors and dual instantaneous Pfaff vectors of dual parallel equidistant ruled surfaces (DPERS) and we give the relationships between these curvatures. In addition to—in cases where the base curves of these DPERS are closed—computing the dual integral invariants of the indicatrix curves. Additionally, we show the relationships between them. Finally, we provide an example for each of these indicatrix curves.

Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces

“…He categorizes the ruled and the translation surfaces [19]. Some authors give crucial results about the curves and surfaces in different spaces [12,15,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. We can find more motivations of our work from some articles (see [38][39][40][41][42][43][44][45][46][47][48][49][50]).…”

Estimation of Ricci Curvature for Hemi-Slant Warped Product Submanifolds of Generalized Complex Space Forms and Their Applications

“…The characterization of differential equations on Riemannian manifolds has grown to be an exciting area of study and has been looked into by many scholars, including these in [24][25][26]. Furthermore, we will combine the methods and results in [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] for future research to obtain more new developments. Note: Throughout the paper, we used the abbreviations w p for "warped product", g.c.s.f for "generalized complex space form", TG for "totally geodesic", and TU for "totally umbilical".…”