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 $\mathcal{A}=(\langle A^{\alpha},A^{\beta}\rangle)_{m\times m}$. We show that if $\lambda_{1}\leqslant n$ and $\rho^{\perp}\leqslant \left[{\sqrt{2}n(n-1)}\right]^{-1} \mathop{\inf}\limits_{p\in M}(n-\lambda_{1})(p)$, then $\rho^{\perp}\equiv 0$, which means the normal bundle of $M^n$ is flat, and further we give the classification of $M^n$.