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arXiv · 1911.04275

On Totally umbilical surfaces in the warped product $\mathbb{M}(κ)_f\times\mathbb{R}$

Ady Cambraia Jr·Abigail Folha·Carlos Peñafiel
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Abstract

In this article we classify the totally umbilical surfaces which are immersed into a wide class of Riemannian manifolds having a structure of warped product, more precisely, we show that a totally umbilical surface immersed into the warped product $\mathbb{M}(κ)_f\times I$ (here, $\mathbb{M}(κ)$ denotes the 2-dimensional space form, having constant curvature $κ$, $I$ an interval and $f$ the warping function) is invariant by an 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 (we mean, the curve which generates a invariant totally umbilical surface). Moreover, we construct explicit examples of totally umbilical surfaces, invariant by one-parameter group of isometries of the ambient space, by considering certain non-trivial warping function.

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Ady Cambraia Jr, Abigail Folha, Carlos Peñafiel. 2020-10-13. On Totally umbilical surfaces in the warped product $\mathbb{M}(κ)_f\times\mathbb{R}$. https://arxiv.org/abs/1911.04275

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