# zbMATH — the first resource for mathematics

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