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On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space. (English) Zbl 0855.53010

The author classifies spacelike and timelike ruled surfaces in 3-dimensional Minkowski spacetime \(\mathbb{R}^3_1\) whose Gauss map \(\xi\) satisfies \(\Delta \xi= A\xi\) for some \(3\times 3\) real matrix \(A\), where \(\Delta\) is the Laplacian of the surfaces. He proves that such surfaces are locally either \(\mathbb{R}^2_1\), \(S^1_1\times \mathbb{R}^1\), \(\mathbb{R}^1_1\times S^1\), \(\mathbb{R}^2\) or \(H^1\times \mathbb{R}^1\). Ruled surfaces in a Euclidean 3-space satisfying \(\Delta\xi= A\xi\) were classified by C. Baikoussis and D. E. Blair in [Glasgow Math. J. 34, No. 3, 355-359 (1992; Zbl 0762.53004)]. Ruled surfaces of finite type in a Euclidean \(n\)-space in general were completely classified by C. Baikoussis, the reviewer and L. Verstraelen in [Tokyo J. Math. 16, No. 2, 341-349 (1993; Zbl 0798.53055)].

MSC:

53B25 Local submanifolds
53C40 Global submanifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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