Abstract
This paper is concerned with the classifications of quadric surfaces of first and second kinds in Euclidean 3-space satisfying the Jacobi condition with respect to their curvatures, the Gaussian curvature K, the mean curvature H, second mean curvature H-II and second Gaussian curvature K-II. Also, we study the zero and non-zero constant curvatures of these surfaces. Furthermore, we investigated the (A, B)-Weingarten, (A, B)-linear Weingarten as well as some special (C-2, K) and (C-2, K root/k)-nonlinear Weingarten quadric surfaces in E-3, where Lambda not equal B, Lambda, B is an element of {K, H, H-II, K-II} and C is an element of {H, H-II, K-II}. Finally, some important new lemmas are presented.