Submanifolds of products of space forms

TL;DR

This paper classifies submanifolds with parallel second fundamental form in products of two space forms and reduces the problem of classifying umbilical submanifolds to known cases, providing new reduction theorems.

Contribution

It provides a complete classification of certain submanifolds in product space forms and introduces reduction theorems for codimension, advancing understanding of submanifold geometry.

Findings

Complete classification of submanifolds with parallel second fundamental form.

Reduction of umbilical submanifold classification to known cases.

Development of codimension reduction theorems for product 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.