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Abstract
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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
(
denotes the 2-dimensional space form having constant curvature
,
is an
interval and
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.
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Keywords
classification theorem, invariant surfaces, cylinders
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Mathematical Subject Classification
Primary: 53C30, 53C42
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Milestones
Received: 21 June 2020
Revised: 5 July 2021
Accepted: 5 July 2021
Published: 12 October 2021
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