Two theorems on invariant submanifolds of product Riemannian manifold. (Q1411771)

Let \(M\) be an invariant submanifold in the sense of [\textit{S. Xu} and \textit{Y. Ni}, Acta Math. Sci. (Engl. Ed.) 20, No. 2, 213--218 (2000; Zbl 0979.53065)] of a product Riemannian manifold \((\overline M_1\times \overline M_2,\overline g)\) and \(f\) a Riemannian almost-product structure on \(M\) with \(f^2= I\). The vertical and horizontal distributions are denoted by \(T_1= X\in TM\mid fX= X\) and \(T_2= X\in TM| fX=-X\), respectively, and their integral manifolds by \(M_1\) and \(M_2\), respectively. It is shown that then \(M\) is a curvature-invariant (resp. pseudo-umbilical) submanifold of \((\overline M_1\times \overline M_2,g)\) if and only if \(M_1\) and \(M_2\) are curvature-invariant (resp. pseudo-umbilical) submanifolds of Riemannian manifolds \((\overline M_1, \overline g_1)\) and \((M_2,\overline g_2)\), respectively. Here, curvature-invariant means that \((\overline\nabla_X h)(Y,Z)= (\overline\nabla_Y h)(X,Z)\) for tangent vector fields \(X\), \(Y\), and \(Z\). If \((\overline M_1\times\overline M_2,\overline g)\) has constant sectional curvature then here a totally umbilical \(M\) is always totally geodesic.