zbMATH — the first resource for mathematics

Rotational hypersurfaces of space forms with constant scalar curvature. (English) Zbl 0695.53040
We denote by \(N_ c\) the simply connected n-dimensional space form of constant curvature \(c=0,1\) or -1. Let M be a complete rotational hypersurface of \(N_ c\) with constant scalar curvature S. In this interesting, clearly written paper the author classifies these hypersurfaces in the cases \(c=0,-1\) and presents partial results for \(c=1\). Moreover he determines the admissible values of S in each of the three cases and gives a geometrical description of the hypersurfaces according to the values of S. In particular he proves that S is precisely greater than or equal to the space form curvature, except in the case \(c=1\) where any value greater than (n-3)/(n-1) is admissible. Surprising examples of embedded hypersurfaces in the case \(c=1\) with \(S<1\) are presented, which are not isometric to a product of spheres.
Reviewer: T.Hasanis

53C40 Global submanifolds
Full Text: DOI EuDML
[1] J. L. BARBOSA and M. do CARMO: Helicoids, catenoids and minimal hypersurfaces ofR n invariant by an -parameter group of motions, An. Acad. Bras. Cienc.53, 403–408 (1981) · Zbl 0493.53044
[2] F. BRITO and M. L. LEITE: Hipersuperficies rotacionais deR n com curvatura escalar constante, Atas do 16o Colóquio Brasileiro de Matemática, Rio de Janeiro (1987)
[3] S. Y. CHENG and S. T. YAU: Hypersurfaces with constant scalar curvature, Math. Ann.225, 195–204 (1977) · Zbl 0349.53041 · doi:10.1007/BF01425237
[4] W. Y. HSIANG: On rotational W-hypersurfaces in spaces of constant curvature and generalized laws of sine and cosine, Bull. Inst. Math. Acad. Sinica11, 349–373 (1983) · Zbl 0525.53059
[5] W. Y. HSIANG: OnW-hypersurfaces of generalized rotational type inE n+1, I, preprint (1982)
[6] E. KASNER: Finite representation of the solar gravitational field in flat space of six dimensions, Am. J. Math.43, 130–133 (1921) · JFM 48.1040.01 · doi:10.2307/2370246
[7] A. ROS: Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Diff. Geom.27, 215–220 (1988) · Zbl 0638.53051
[8] M. SPIVAK: A comprehensive introduction to Differential Geometry, vol 4,2nd edn, Publish or Perish, Inc. Berkeley (1979) · Zbl 0439.53003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.