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Journal of the London Mathematical Society 2001 64(3):755-768; doi:10.1112/S0024610701002587
© 2001 by London Mathematical Society
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© The London Mathematical Society

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

Qing-Ming Cheng

Department of Mathematics, Faculty of Science and Engineering, Saga University Saga 840-8502, Japan, cheng{at}ms.saga-u.ac.jp

Received 2 November 2000. Revision received 12 March 2001.

The paper considers n-dimensional hypersurfaces with constant scalar curvature of a unit sphere Sn–1(1). The hypersurface Sk(c1)xSn–k(c2) in a unit sphere Sn+1(1) is characterized, and it is shown that there exist many compact hypersurfaces with constant scalar curvature in a unit sphere Sn+1(1) which are not congruent to each other in it. In particular, it is proved that if M is an n-dimensional (n > 3) complete locally conformally flat hypersurface with constant scalar curvature n(n–1)r in a unit sphere Sn+1(1), then r > 1–2/n, and

(1) when r != (n–2)/(n–1), if


Formula

then M is isometric to S1FormulaxSn–1(c), where S is the squared norm of the second fundamental form of M;

(2) there are no complete hypersurfaces in Sn+1(1) with constant scalar curvature n(n–1)r and with two distinct principal curvatures, one of which is simple, such that r = (n–2)/(n–1) and


Formula


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