Abstract
We provide a survey of applications of Simons' type for- mula to submanifolds with constant mean curvature or with parallel mean curvature vector in Riemannian space forms. Also, we show a result of reduction of codimension for complete submanifolds such that the normalized mean curvature vector is parallel and the squared norm of the second fundamental form satises certain inequality. At the end, we give some open questions to submanifolds in general products of Riemannian space forms.
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