#### Vol. 313, No. 2, 2021

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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 $\mathbb{𝕄}{\left(\kappa \right)}_{f}×I$ ($\mathbb{𝕄}\left(\kappa \right)$ 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
##### Authors
 Ady Cambraia Jr. Departamento de Matemática Universidade Federal de Viçosa Minas Gerais Brazil Abigail Folha Departamento de Geometria Universidade Federal Fluminense Rio de Janeiro Brazil Carlos Peñafiel Instituto de Matemática Universidade Federal de Rio de Janeiro Rio de Janeiro Brazil