Totally real submanifolds in a quaternion projective space (Q675767)

Let \(M\) be an \(n\)-dimensional compact totally real minimal submanifold in the quaternionic projective space \(QP^n(c)\) of constant quaternionic sectional curvature \(c\). Denote by \(\rho\) the scalar curvature of \(M\), by \(\sigma\) the second fundamental form of \(M\), and by \(K_c\) and \(Q\) the functions assigning to each point \(p\in M\) the infimum of the sectional curvature and Ricci curvature of \(M\) at \(p\), respectively. Suppose that one of the following inequalities holds: (1) \(|\sigma|^2 \leq (n+1)c/6\), (2) \(\rho\geq(3n^2-5n-2)c/12\), (3) \(K_c\geq (2n-3)c/20n\), (4) \(Q\geq (2n-3)c/10\). Then \(M\) is either totally geodesic in \(QP^n(c)\) or (only in case (1) or (2)) \(n=2\) and \(M\) is a flat surface in \(QP^2(c)\) with parallel second fundamental form and parallel normal bundle. These results improve previous ones by \textit{Y.-B. Shen} [Chin. Ann. Math., Ser. B 14, 297-306 (1993; Zbl 0811.53061)].