ON SOME GEOMETRIC PROPERTIES OF QUADRIC SURFACES IN EUCLIDEAN SPACE

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 <TEX>$H_{II}$</TEX> and second Gaussian curvature <TEX>$K_{II}$</TEX>. 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 (<TEX>$C^2$</TEX>, K) and <TEX>$(C^2,\;K{\sqrt{K}})$</TEX>-nonlinear Weingarten quadric surfaces in <TEX>$E^3$</TEX>, where <TEX>$A{\neq}B$</TEX>, A, <TEX>$B{\in}{K,H,H_{II},K_{II}}$</TEX> and <TEX>$C{\in}{H,H_{II},K_{II}}$</TEX>. Finally, some important new lemmas are presented.