Pinching rigidity theorems for normal scalar curvature

Let $M^n$ be an $n$-dimensional closed minimal submanifold immersed in the unit sphere ${\mathbb{S}}^{n+m}$. Denote by $S$ and ${\rho }^{\perp }$ the squared norm of the second fundamental form and the normal scalar curvature of $M^n$, respectively. Let $\{A^{\alpha }{\}}_{\alpha =n+1}^{n+m}$ be the shape operators of $M^n$ with respect to a local orthonormal normal frame. Denote by ${\lambda }_1$ the largest eigenvalue of the positive semi-definite symmetric matrix $\mathcal{A}=(⟨A^{\alpha },A^{\beta }⟩{)}_{m\times m}$. We show that if ${\lambda }_1⩽n$ and ${\rho }^{\perp }⩽{\left[\sqrt{2}n(n-1)\right]}^{-1}\underset{p\in M}{}(n-{\lambda }_1)(p)$, then ${\rho }^{\perp }\equiv 0$, which means the normal bundle of $M^n$ is flat, and further we give the classification of $M^n$.