Abstract. In this paper, we study surfaces of revolution without parabolic points in 3-Euclidean space R 3 , satisfying the condition ∆ II G = f (G + C), where ∆ 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 R 3 which satisfy the condition ∆ II G = f G, coincide with surfaces of revolution with non-zero constant Gaussian curvature.

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In this paper, we study helicoidal surfaces without parabolic points in Euclidean 3-space R 3 , satisfying the condition Δ II G = f (G + C), where Δ 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 helicoidal surfaces without parabolic points in R 3 which satisfy the condition Δ II G = f (G + C), coincide with helicoidal surfaces with non-zero constant Gaussian curvature.Mathematics Subject Classification. Primary 53A05; Secondary 53B25, 53C40.