Submanifolds of products of space forms

We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension $m\ge 3$ of a product ${\text{Q}}_{k_1}^{n_1}\times {\text{Q}}_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2\mathrm{\ne }0$ to the classification of $m$-dimensional umbilical submanifolds of codimension two of ${\text{Sf}}^n\times \mathbb{R}$ and ${\text{Hy}}^n\times \mathbb{R}$. The case of ${\text{Sf}}^n\times \mathbb{R}$ was carried out in \cite{mt}. As a main tool we derive reduction of codimension theorems of independent interest for submanifolds of products of two space forms.