SURFACES OF REVOLUTION SATISFYING ΔIIG = f(G + C)

Abstract

In this paper, we study surfaces of revolution without parabolic points in 3-Euclidean space <TEX>$\mathbb{R}^3$</TEX>, satisfying the condition <TEX>${\Delta}^{II}G=f(G+C)$</TEX>, where <TEX>${\Delta}^{II}$</TEX> is the Laplace operator with respect to the second fundamental form, <TEX>$f$</TEX> is a smooth function on the surface and C is a constant vector. Our main results state that surfaces of revolution without parabolic points in <TEX>$\mathbb{R}^3$</TEX> which satisfy the condition <TEX>${\Delta}^{II}G=fG$</TEX>, coincide with surfaces of revolution with non-zero constant Gaussian curvature.