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

Let $M^n\subset S^{n+m}$ be a compact minimal submanifold of the unit sphere, and let ${\lambda }_1\ge \cdots \ge {\lambda }_m\ge 0$ be the eigenvalues of its fundamental matrix. We prove that ${\sum }_{\alpha =1}^{\min \{n,m\}}{\lambda }_{\alpha }+{\lambda }_2\le n$ forces $M$ to be a totally geodesic subsphere, a generalized Clifford torus, or a Veronese manifold. We first show that equality everywhere in the pinching condition forces the second fundamental form to be parallel. Next, by computing the eigenvalues of the fundamental matrices of irreducible symmetric $R$-spaces, we obtain a complete classification of the equality cases among minimal submanifolds with parallel second fundamental form.