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The family of cones is one of typical models of non-cylindrical ruled surfaces. Among them, the circular cones are unique in the sense that their Gauss map satisfies a partial differential equation similar, though not identical, to one characterizing the so-called 1-type submanifolds. Specifically, for the Gauss map G of a circular cone, one has ∆G = f (G + C), where ∆ is the Laplacian operator, f is a non-zero function and C is a constant vector. We prove that circular cones are characterized by being the only non-cylindrical ruled surfaces with ∆G = f (G + C) for a nonzero constant vector C.

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In this paper, we prove two optimal inequalities involving the intrinsic scalar curvature and extrinsic Casorati curvature of submanifolds of real space forms endowed with a semi-symmetric metric connection. Moreover, we show that in both cases, the equality at all points characterizes the invariantly quasi-umbilical submanifolds. MSC: 53C40; 53B05

In Theorem 3.4 in [1], we considered tube in the 3-dimensional Euclidean space E 3 satisfying the linear equation aK + bH = c for a, b, c ∈ R, where K and H denote the Gaussian curvature and the mean curvature, respectively.We found some mistakes on the statement and the proof of Theorem 3.4. In fact, the statement and the proof of Proof. Let T r (γ) be a tube parametrized by x = x(t, θ) = γ(t) + r(cos θn(t) + sin θb(t)),where n and b are the principal normal vector and the binormal vector of a smooth unit speed curve γ. Then the Gaussian curvature K and the mean curvature H in [1] are given bywhere α = 1 − rκ(t) cos θ. In this case, the mean curvature H can be written as H = 1 2rα − r rα κ cos θ = 1 2rα + rK, which implies