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
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