Totally Umbilical Hypersurfaces of Product Spaces

2021/01/03 by de Lima, Ronaldo F., Santos, João Paulo dos · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2101.00634

Abstract

Given a Riemannian manifold M, and an open interval I⊂ℝ, we characterize nontrivial totally umbilical hypersurfaces of the product M× I -- as well as of warped products I×_ωM -- as those which are local graphs built on isoparametric families of totally umbilical hypersurfaces of M. By means of this characterization, we fully extend to \mathbbSⁿ×ℝ and ℍⁿ×ℝ the results by Souam and Toubiana on the classification of totally umbilical hypersurfaces of \mathbbS²×ℝ and ℍ²×ℝ. It is also shown that an analogous classification holds for arbitrary warped products I×_ω\mathbbSⁿ and I×_ωℍⁿ.

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