| On Einsteinian Manifolds Admitting Orthogonal Families of Totally Umbilical Hypersurfaces |
| Received:November 18, 1981 |
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| Abstract: |
| The aim of the present paper is to study globally the Riemannian manifold admitting two or more mutually orthogonal families of totally umbilical hypersurfaccs of which each is Einsteinian. This paper consists of four parts: (i) to establish anew the canonical form of the metric of (M,g) admitting p (p≥2) families of mutually orthogonal totally umbilical hypcrsurf aces from the standpoint of global differential geometry; (ii) to prove in a n-dimensional (n>2) Einsteinian manifold En of nonvanishing scalar curvature there doesn't exist one family of compact totally geodesic Einsteinian hypersurfaces (Theorem 1);(iii) to prove in a n-dimensional (n≥5) Einsteinian manifold En of nonnegative scalar curvature R there don't exist two orthogonal families of totally umbilical but not geodesic complete Einsteinian hypersurfaces (Theorem Ⅱ);(iv) to show that a n-dimensional (n≥5) Riemannian manifold of negative constant scalar curvature R. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.1983.03.002 |
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