On totally umbilical surfaces in the warped product $\mathbb{M}(\kappa)_f\times I$

Ady Cambraia Jr., Abigail Folha and Carlos Peñafiel

Vol. 313 (2021), No. 2, 343–364

DOI: 10.2140/pjm.2021.313.343

Abstract

We classify the totally umbilical surfaces which are immersed into a wide class of Riemannian manifolds having the structure of a warped product; more precisely, we show that a totally umbilical surface immersed into the warped product $\mathbb{𝕄}{\left(\kappa \right)}_f\times I$ ($\mathbb{𝕄}\left(\kappa \right)$ denotes the 2-dimensional space form having constant curvature $\kappa$, $I$ is an interval and $f$ is the warping function) is invariant by a one-parameter group of isometries of the ambient space. We also find the first integral of the ordinary differential equation that the profile curve satisfies (i.e., the curve which generates an invariant totally umbilical surface). Moreover, we construct explicit examples of totally umbilical surfaces, invariant by one-parameter groups of isometries of the ambient space, by considering a certain nontrivial warping function.

Keywords

classification theorem, invariant surfaces, cylinders

Mathematical Subject Classification

Primary: 53C30, 53C42

Milestones

Received: 21 June 2020

Revised: 5 July 2021

Accepted: 5 July 2021

Published: 12 October 2021