An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices

Based on an inequality on the upper bound of the sum of squared norms of Lie bracket of symmetric matrices, we establish a rigidity theorem for compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices, which are the critical values of the squared norms of the second fundamental form on all normal directions. This conclusion is optimal for all dimensions without the restriction of the codimension, giving a new characterization for generalized Clifford tori and Veronese manifolds.