article

Title: Submanifolds of products of space forms

Authors: Bruno Mendonca and Ruy Tojeiro

Issue: Volume 62 (2013), Issue 4, 1283-1314

Abstract:

$We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also address the classification of umbilical submanifolds with dimension $m\geq3$ of a product $\mathbb{Q}_{k_1}^{n_1}\times\mathbb{Q}_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2\neq0$. This is reduced to the classification of $m$-di\-men\-sional umbilical submanifolds of codimension two of $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$. The case of $\mathbb{S}^n\times\mathbb{R}$ was carried out in [B. Mendon\c a and R. Tojeiro, \textit{Umbilical submanifolds of $\mathbb{S}^n\times\mathbb{R}$}, preprint, available at http://arxiv.org/abs/arXiv1107.1679.v2[mathDG]]. As a main tool, we derive reduction of codimension theorems of independent interest for submanifolds of products of two space forms.$