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On hypersurfaces in hyperquadrics. (English) Zbl 0789.53037
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 469-479 (1992).
Let \(\mathbb{R}^{n+1}_ \nu\) be the semi-Euclidean space and put \(S^ n_ \nu(r) = \{x\in \mathbb{R}^{n+1}_ \nu; | \langle P(x),P(x)\rangle| = r^ 2\}\), where \(P(x)\) denotes the position vector field. The connected components of \(S^ n_ \nu(r)\) are called hyperquadrics and each hyperquadric carries a fixed indicator \(\varepsilon_ P = \text{sign}\langle P,P\rangle\). The main result of the present paper is as follows. Let \(M\) be a geodesically complete, connected semi-Riemannian hypersurface in \(S^ n_ \nu\) with position indicator \(\varepsilon_ P\). Suppose that there exists a coordinate function \(x^ i\) such that \(\Delta_ M x^ i = -(n-1)\varepsilon_ P\cdot x^ i\). Then either \(M\) has everywhere vanishing mean curvature or else \(M\) contains strips of generalized cylinders in \(S^ n_ \nu(r)\) with geodesic generators orthogonal to the totally geodesic hypersurface \(V_ i = S^ n_ \nu \cup \{x\in R^{n+1}_ \nu; x^ i = 0\}\).
For the entire collection see [Zbl 0764.00002].
MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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