Hypersurfaces With Constant Mean Curvature In A Riemann Manifold With Constant Curvature And With Quasi Constant Curvature

In this paper we study hypersurfaces with constant mean curvature in a Riemann manifold with constant curvature and with quasi constant curvature by the use of moving frames by Elie Cartan and get two sufficient conditions that a hypersurface M be a totally geodesic hypersurface, in the same time we derive three corollaries from one of the theorems obtained. The main results obtained in the present paper are that 1° Let M be a compact oriented hypersurface with constant mean curvature H in a Riemann manifold Nⁿ⁺¹(c) of constant curvature. If and the components of the Ricci tensor for Mare then Mis totally geodesic. 2° Let M be a compact oriented hypersurface with constant mean curvature H in a Riemann manifold Nⁿ⁺¹ of quasi constant curvature, and assume that and the components of the Ricci tensor for M are everywhere on M, then Mis totally geodesic.