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Register a new userEuclidean hypersurfaces with a totally geodesic foliation of codimension one
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We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also considered.
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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 given in cite{drt} 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.
In this paper we classify Euclidean hypersurfaces $fcolon M^n rightarrow mathbb{R}^{n+1}$ with a principal curvature of multiplicity $n-2$ that admit a genuine conformal deformation $tilde{f}colon M^n rightarrow mathbb{R}^{n+2}$. That $tilde{f}colon M^n rightarrow mathbb{R}^{n+2}$ is a genuine conformal deformation of $f$ means that it is a conformal immersion for which there exists no open subset $U subset M^n$ such that the restriction $tilde{f}|_U$ is a composition $tilde f|_U=hcirc f|_U$ of $f|_U$ with a conformal immersion $hcolon Vto mathbb{R}^{n+2}$ of an open subset $Vsubset mathbb{R}^{n+1}$ containing $f(U)$.
In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem.
In the realm of conformal geometry, we give a classification of the Euclidean hypersurfaces that admit a non-trivial conformal infinitesimal variation. In the restricted case of conformal variations, such a classification was obtained by E. Cartan in 1917. The case of infinitesimal isometric variations was done by U. Sbrana in 1908. In particular, we show that the class of hypersurfaces that allow a conformal infinitesimal variation is much larger than the one considered by Cartan.
The main purpose of this paper is to complete the work initiated by Sbrana in 1909 giving a complete local classification of the nonflat infinitesimally bendable hypersurfaces in Euclidean space.
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