Bulletin of the
Korean Mathematical Society BKMS

Involving the Ricci curvature and the squared mean curvature, we obtain a basic inequality for an integral submanifold of an $\mathcal{S}$-space form. By polarization, we get a basic inequality for Ricci tensor also. Equality cases are also discussed. By giving a very simple proof we show that if an integral submanifold of maximum dimension of an $\mathcal{S}$-space form satisfies the equality case, then it must be minimal. These results are applied to get corresponding results for $C$-totally real submanifolds of a Sasakian space form and for totally real submanifolds of a complex space form.

Keywords: $\mathcal{S}$-space form, integral submanifold, $C$-totally real submanifold, totally real submanifold, Lagrangian submanifold, Ricci curvature, $k$-Ricci curvature, scalar curvature.

MSC numbers: Primary 53C40, 53C15, 53C25