Abstract: 1The complete minimal hypersurfaces in a Locally symmetric manifold Nⁿ⁺¹ are studied, with some characteristics of these hypersurfaces obtained by using the generalized maximal principle. It is shown that if M is a complete minimal hypersurface in Nⁿ⁺¹, then M is totally geodesic or sup S is not less than (2δ-1)n. And it is shown furture that M is totally geodesic or is the product of Riemannian Manifold of m dimensional and n-m dimensional, whose constant sectional curvature is n/m and m/(n-m), respectively; or sup S is larger than (2δ-1)n. These results generalize the result of Shui N.X. and improve the result of Hineva S.

Key words: locally symmetric, hypersurface, totally geodesic