Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature |
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Authors: | Cheng Qing-Ming |
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Institution: | Department of Mathematics, Faculty of Science and Engineering, Saga University Saga 840-8502, Japan, cheng{at}ms.saga-u.ac.jp |
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Abstract: | The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere Sn1(1). The hypersurfaceSk(c1)xSnk(c2) in a unit sphere Sn+1(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sn+1(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n1)r in a unit sphere Sn+1(1), then r > 12/n,and (1) when r (n2)/(n1), if
then M is isometric to S1xSn1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sn+1(1) with constantscalar curvature n(n1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n2)/(n1)and
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