The paper considers n-dimensional hypersurfaces with constant scalar curvature of a unit sphere Sn−1(1).
The hypersurface Sk(c1)×Sn−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
then M is isometric to S1(√1−c2)×Sn−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