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 Pe├▒afiel

Vol. 313 (2021), No. 2, 343ÔÇô364
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 𝕄(κ)f × I (𝕄(κ) denotes the 2-dimensional space form having constant curvature κ, 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 Vi├žosa
Minas Gerais
Brazil
Abigail Folha
Departamento de Geometria
Universidade Federal Fluminense
Rio de Janeiro
Brazil
Carlos Pe├▒afiel
Instituto de Matemática
Universidade Federal de Rio de Janeiro
Rio de Janeiro
Brazil