In this note, we investigate the pinching problem for oriented compact submanifolds of dimension n with parallel normalized mean curvature vector in a space form Fn+p(c). We first prove a codimension reduction theorem for submanifolds under lower Ricci curvature bounds. Moreover, if the submanifolds have constant normalized scalar curvature R≥c, we obtain a classification theorem for submanifolds under lower Ricci curvature bounds. It should be emphasized that our Ricci pinching conditions are sharp for even n and p=2.
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