Abstract:  Let M be a compact hypersurface in an(n+1)-dimensional complete constant curvature space N(c).If Ricci curvature of M is not less than max{0,(n-1)c}and there is a constant main curvature function in M,then M can be classified completly.This is the Liebmann theorem in the widest sense so far.The methods used in this paper can be used to generalize a class of theorems with non-negative(or positive)sectional curvature conditions.

Key words:  , hypersurfaces, main curvature, symmetric functions, Liebmann theorem