Submanifolds of products of space forms

2012/03/16 by Mendonça, Bruno, Tojeiro, Ruy · 1 citation
#53B25 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1203.3790

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≥ 3 of a product \Q_k1^n₁× \Q_k2^n₂ of two space forms whose curvatures satisfy k₁+k₂≠ 0 to the classification of m-dimensional umbilical submanifolds of codimension two of \Sfⁿ× \R and \Hyⁿ× \R. The case of \Sfⁿ× \R was carried out in \citemt. As a main tool we derive reduction of codimension theorems of independent interest for submanifolds of products of two space forms.

Cited by

• On FKM isoparametric hypersurfaces in \mathbbSⁿ × \mathbbSⁿ and new area-minimizing cones

Related

• Umbilical surfaces of products of space forms
• Umbilical submanifolds of ℍ^k× \mathbbS^n-k+1
• Umbilical submanifolds of \mathbbSⁿ× ℝ
• Isometric immersions into products of many space forms: an introductory study
• On complete submanifolds with parallel mean curvature in product spaces