Curvature pinching for three-dimensional submanifolds in a Riemannian manifold

Let $M^3$ be an oriented submanifold with parallel mean curvature vector in a complete simply connected Riemannian manifold $N^{3+p}$. When the mean curvature $H=0$, i.e., M is minimal, we prove that there exists a constant ${\delta }_1\left(p\right)\in (0,1)$, such that if ${\bar{K}}_N\in \left[{\delta }_1\right(p),1]$, and if M has a lower bound for Ricci curvature and an upper bound for scalar curvature, then $N^{3+p}$ is isometric to $S^{3+p}$. Moreover, M is the totally geodesic sphere $S^3$. This is a generalization of Shen and Li's results [10], [14]. When the ambient manifold is a space form, we improve the geometric rigidity theorem due to Xu-Gu [19] for the codimension is not more than 2 and $H\ne 0$.