Euclidean hypersurfaces with a totally geodesic foliation of codimension one

Dajczer, Marcos; Rovenski, Vladimir; Tojeiro, Ruy

doi:10.1007/s10711-014-9964-4

“…But this implies that ψ(U) is a surfacelike strip, which is ruled out by assumption. The proof then proceeds as in [6] by showing that this implies ψ to be either ruled or a partial tube over a curve.…”

Section: The Case Of Euclidean Hypersurfacesmentioning

confidence: 96%

“…

In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the leaves of the foliation are complete was already given in [6] for Euclidean hypersurfaces. We prove that there exists exactly one further class of local examples in Euclidean space, all of which have rank two.

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mentioning

confidence: 99%

“…In [6] the following basic problem was addressed: Which are the Euclidean hypersurfaces of dimension at least three that carry a totally geodesic foliation of codimension one? Recall that a smooth foliation F on a Riemannian manifold M n is totally geodesic if all leaves of F are totally geodesic submanifolds of M n , that is, if any geodesic of M n that is tangent to F at some point is (locally) contained in the leaf of F through that point.…”

mentioning

confidence: 99%

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Hypersurfaces of space forms carrying a totally geodesic foliation

Dajczer

¹

,

Tojeiro

²

2019

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In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the leaves of the foliation are complete was already given in [6] for Euclidean hypersurfaces. We prove that there exists exactly one further class of local examples in Euclidean space, all of which have rank two. We also extend the classification under the global assumption of completeness of the leaves for hypersurfaces of the sphere and show that there exist plenty of examples in hyperbolic space.

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“…In [5] and [6], the authors studied the problem of determining the hypersurfaces of dimension at least three of a space form that carry a totally geodesic foliation of codimension one. The initial motivation of this work was to investigate the similar problem that one can pose by assuming the foliation to be spherical instead of being totally geodesic.…”

Section: Introductionmentioning

confidence: 99%

“…In the solution of the problem addressed in [5] and [6], one main example of a hypersurface of a space form that carries a totally geodesic foliation of codimension one is a partial tube over a smooth regular curve. Partial tubes are submanifolds that are generated by starting with any submanifold whose normal bundle has a parallel and flat subbundle, taking a submanifold in the fiber of that subbundle at a given point, and then parallel transporting it along the former submanifold with respect to its normal connection.…”

Section: Introductionmentioning

confidence: 99%

Ribaucour partial tubes and hypersurfaces of Enneper type

Chion,

Tojeiro

2021

Preprint

Self Cite

1

0

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In this article we introduce the notion of a Ribaucour partial tube and use it to derive several applications. These are based on a characterization of Ribaucour partial tubes as the immersions of a product of two manifolds into a space form such that the distributions given by the tangent spaces of the factors are orthogonal to each other with respect to the induced metric, are invariant under all shape operators, and one of them is spherical. Our first application is a classification of all hypersurfaces with dimension at least three of a space form that carry a spherical foliation of codimension one, extending previous results by Dajczer, Rovenski and the second author for the totally geodesic case. We proceed to prove a general decomposition theorem for immersions of product manifolds, which extends several related results. Other main applications concern the class of hypersurfaces of R n+1 that are of Enneper type, that is, hypersurfaces that carry a family of lines of curvature, correspondent to a simple principal curvature, whose orthogonal (n − 1)-dimensional distribution is integrable and whose leaves are contained in hyperspheres or affine hyperplanes of R n+1 . We show how Ribaucour partial tubes in the sphere can be used to describe all n-dimensional hypersurfaces of Enneper type for which the leaves of the (n − 1)-dimensional distribution are contained in affine hyperplanes of R n+1 , and then show how a general hypersurface of Enneper type can be constructed in terms of a hypersurface in the latter class. We give an explicit description of some special hypersurfaces of Enneper type, among which are natural generalizations of the so called Joachimsthal surfaces.

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