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Abstract

Let $M$ be an $n$-dimensional complete immersed submanifold with parallel mean curvature vectors in an $(n+p)$-dimensional Riemannian manifold $N$ of constant curvature $c>0$. Denote the square of length and the length of the trace of the second fundamental tensor of $M$ by $S$ and $H$, respectively. We prove that if $$ S\leq\frac{1}{n-1}\,H^2+2c,\ \quad n\geq 4, $$ or $$ S\leq\frac{1}{2} \, H^2 + \min\bigg(2,\frac{3p-3}{2p-3}\bigg)c,\ \quad n=3, $$ then $M$ is umbilical. This result generalizes the Okumura–Hasanis pinching theorem to the case of higher codimensions.