Kaehler-submanifolds in a locally symmetric Bochner-Kaehler manifold (Q1826149)

Using the maximum principle the authors prove the following results which are generalizations of results of \textit{A. Ros} [Proc. Am. Math. Soc. 93, 329-331 (1985; Zbl 0561.53055)] and \textit{A. Ros} and \textit{L. Verstraelen} [J. Differ. Geom. 19, 561-566 (1984; Zbl 0535.53045)]: Theorem 1. Let \(M^ n\) be a compact Kaehler submanifold of complex dimension \(n\geq 2\) in a locally symmetric Bochner-Kaehler manifold \(\tilde M^{n+p}\) of complex dimension \(n+p\). Denote \[ Q_{\max}=\max_{x\in M^ n}\{Ric(\tilde M^{n+p})_ x\}\quad and\quad Q_{\min}=\min_{x\in M^ n}\{Ric(\tilde M^{n+p})_ x\}. \] If the sectional curvature \(K(M^ n)\) of \(M^ n\) satisfies \[ K(M^ n)>\frac{1}{2(n+p+2)}[Q_{\max}-\frac{n+p}{2(n+p+1)}Q_{\min}], \] then \(M^ n\) must be totally geodesic. Theorem 2. Let \(M^ n\) and \(M^{n+p}\) be the same as in Theorem 1. If the holomorphic sectional curvature \(H(M^ n)\) of \(M^ n\) satisfies \[ H(M^ n)>\frac{2}{n+p+2}[Q_{\max}-\frac{n+p}{2(n+p+1)}Q_{\min}]\quad, \] then \(M^ n\) must be totally geodesic.