On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms

Let M n be a Riemannian n-manifold. Denote by S(p) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\overline {Ric}(p)$\end{document} the Ricci tensor and the maximum Ricci curvature on M n at a point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $p\in M^n$\end{document}, respectively. First we show that every isotropic submanifold of a complex space form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\widetilde M^m(4\,c)$\end{document} satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $S\leq ((n-1)c+ {n^2 \over 4} H^2)g$\end{document}, where H2 and g are the squared mean curvature function and the metric tensor on M n, respectively. The equality case of the above inequality holds identically if and only if either M n is totally geodesic submanifold or n = 2 and M n is a totally umbilical submanifold. Then we prove that if a Lagrangian submanifold of a complex space form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\widetilde M^m(4\,c)$\end{document} satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\overline {Ric}= (n-1)c+ {n^2 \over 4} H^2$\end{document} identically, then it is a minimal submanifold. Finally, we describe the geometry of Lagrangian submanifolds which satisfy the equality under the condition that the dimension of the kernel of second fundamental form is constant.