Submanifolds of products of space forms

Abstract

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 $m\geq 3$ of a product $\Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2\neq 0$ to the classification of $m$-dimensional umbilical submanifolds of codimension two of $\Sf^n\times \R$ and $\Hy^n\times \R$. The case of $\Sf^n\times \R$ 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.