COMPLETE LINEAR WEINGARTEN SPACELIKE SUBMANIFOLDS IMMERSED IN THE ANTI-DE SITTER SPACE

We deal with n-dimensional complete linear Weingarten spacelike submanifolds having nonnegative sectional curvature and immersed in the anti-de Sitter space H-p(n+p) of index p with parallel normalized mean curvature vector field. We recall that a spacelike submanifold is said to be linear Weingarten when its mean and normalized scalar curvature functions are linearly related. We prove that under suitable constraints on the mean curvature function, such a spacelike submanifold must be either totally umbilical or isometric to a product M-1 x . . . x M-k, where the factors M-i are totally umbilical submanifolds of H(p)(n+p )which are mutually perpendicular along their intersections. Furthermore, when this spacelike submanifold is assumed to be compact (without boundary) with positive sectional curvature, we also obtain a rigidity result.