Bulletin of the
Korean Mathematical Society BKMS

In this paper, we study surfaces of revolution without parabolic points in $ 3 $-Euclidean space $ \mathbb{R} ^{3}$, satisfying the condition $\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)$, where $\Delta ^{II}$ is the Laplace operator with respect to the second fundamental form, $f$ 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 $ \mathbb{R} ^{3}$ which satisfy the condition $\Delta ^{II}\mathbf{G}=f\mathbf{G}$, coincide with surfaces of revolution with non-zero constant Gaussian curvature.

Keywords: surfaces of revolution, Laplace operator, pointwise $1$-type Gauss map, second fundamental form

MSC numbers: 53A05, 53B25, 53C40