TAILIEUCHUNG - Weingarten quadric surfaces in a Euclidean 3-space
In this paper, we study quadric surfaces in a Euclidean 3-space. Furthermore, we classify quadric surfaces in a Euclidean 3-space in terms of the Gaussian curvature and the mean curvature. | Turk J Math 35 (2011) , 479 – 485. ¨ ITAK ˙ c TUB doi: Weingarten quadric surfaces in a Euclidean 3-space Min Hee Kim and Dae Won Yoon Abstract In this paper, we study quadric surfaces in a Euclidean 3-space. Furthermore, we classify quadric surfaces in a Euclidean 3-space in terms of the Gaussian curvature and the mean curvature. Key Words: Quadric surface, Weingarten surface, Gaussian curvature, mean curvature. 1. Introduction det A Weingarten surface is a surface on which there exists the Jacobi equation Φ(k1 , k2 ) = (k1 )s (k1 )t = 0 between the principal curvatures k1 , k2 on a surface, or equivalently, the Jacobi (k2 )s (k2 )t equation Ψ(H, K) = 0 between the Gaussian curvature K and the mean curvature H on a surface, where (k1 )s = ∂k1 ∂s and (k2 )t = ∂k2 ∂t . On the other hand, if a surface satisfies a linear equation ak1 + bk2 = c or aK + bH = c for some real numbers a, b, c with (a, b) = (0, 0), then it is said to be a linear Weingarten surface. For the study of these surfaces, W. K¨ uhnel ([5]) investigated ruled Weingarten surface in a Euclidean 3-space E3 . F. Dillen and W. K¨ uhnel ([2]) and Y. H. Kim and D. W. Yoon ([4]) gave a classification of ruled Weingarten surfaces and ruled linear Weingarten surfaces in a Minkowski 3-space E31 , respectively. D. W. Yoon ([10]) classified ruled linear Weingarten surface in E3 . Recently, M. I. Munteanu and I. Nistor ([9]) and R. Lo´ pez ([6, 7]) studied polynomial translation (linear) Weingarten surfaces and a cyclic linear Weingarten surface in a Euclidean 3-space, respectively. In [8] R. Lo´ pez classified all parabolic linear Weingarten surfaces in hyperbolic 3-space. In this paper, we study quadric surfaces in a Euclidean 3-space and prove the following classification theorem. Theorem A. Let M be a quadric surface in a Euclidean 3-space with non-zero Gaussian curvature everywhere. If M satisfies the Jacobi equation with respect to the Gaussian curvature K and the mean .
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