Codimension Reduction of Minimal Submanifolds with Flat Normal Bundle in Spheres

Let $x:M^n\to {\mathbb{S}}^{n+q}$ be a minimal immersion of a complete connected manifold with flat normal bundle, where $n\ge 3$ and $q\ge 2$. We prove that if the squared norm of the second fundamental form $S$ satisfies $S<2n$, then $M^n$ lies in ${\mathbb{S}}^{n+1}$. The standard minimal products of three spheres show that the upper bound $2n$ is optimal. We also prove a codimension reduction theorem under the condition $S+{\lambda }_2\le 2n+\frac{2n}{3n-2}$, where ${\lambda }_2$ is the second largest eigenvalue of the Gram matrix of the shape operators, implying that $M^n$ lies in ${\mathbb{S}}^{n+1}$.