Consider a Riemannian manifold isometrically immersed into a Riemannian manifold of constant sectional curvature such that the mean curvature vector field is parallel, the norm of the second fundamental form is constant, and a certain natural identity is satisfied. Such immersions turn out to be isoparametric hypersurfaces in codimension one, minimal immersions in codimension two, and totally geodesic immersions in higher codimensions. This is proved by developing an inequality that implies that the second fundamental form is parallel for immersions of the type under consideration.