arXiv subjects

Ruy Tojeiro

Publications and source records attributed to Ruy Tojeiro.

27 records · Page 2Linked to original sources

arXiv2009

On a class of hypersurfaces in $\Sf^n\times \R$ and $\Hy^n\times \R$

We give a complete description of all hypersurfaces of the product spaces $\Sf^n\times \R$ and $\Hy^n\times \R$ that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces $\R^{n+2}\supset \Sf^n\times \R$ and $\Le^{n+2}\supset \Hy^n\times \R$. We prove that any such hypersurface in $\Sf^n\times \R$ (respectively, $\Hy^n\times \R$) can be constructed by means of a family of parallel hypersurfaces in $\Sf^n$ (respectively, $\Hy^n$) and a smooth function of one variable. Then we show that constant mean curvature hypersurfaces in this class are given in terms of an isoparametric family in the base space and a solution of a certain ODE. For minimal hypersurfaces such solution is explicitly determined in terms of the mean curvature function of the isoparametric family. As another consequence of our general result, we classify the constant angle hypersurfaces of $\Sf^n\times \R$ and $\Hy^n\times \R$, that is, hypersurfaces with the property that its unit normal vector field makes a constant angle with the unit vector field spanning the second factor $\R$. This extends previous results by Dillen, Fastenakels, Van der Veken, Vrancken and Munteanu for surfaces in $\Sf^2\times \R$ and $\Hy^2\times \R$. Our method also yields a classification of all Euclidean hypersurfaces with the property that the tangent component of a constant vector field in the ambient space is a principal direction, in particular of all Euclidean hypersurfaces whose unit normal vector field makes a constant angle with a fixed direction.

math.DG↗

arXiv2008

Genuine deformations of submanifolds II:the conformal case

We extend to the conformal realm the concept of genuine deformations of submanifolds, introduced by Dajczer and the first author for the isometric case. Analogously to that case, we call a conformal deformation of a submanifold $M^n$ genuine if no open subset of $M^n$ can be included as a submanifold of a higher dimensional conformally deformable submanifold in such a way that the conformal deformation of the former is induced by a conformal deformation of the latter. We describe the geometric structure of a submanifold that admits a genuine conformal deformation and give several applications showing the unifying character of this concept.

math.DG↗

arXiv2007

All superconformal surfaces in \R^4 in terms of minimal surfaces

We give an explicit construction of any simply-connected superconformal surface $ϕ\colon M^2\to \R^4$ in Euclidean space in terms of a pair of conjugate minimal surfaces $g,h\colon M^2\to\R^4$. That $ϕ$ is superconformal means that its ellipse of curvature is a circle at any point. We characterize the pairs $(g,h)$ of conjugate minimal surfaces that give rise to images of holomorphic curves by an inversion in $\R^4$ and to images of superminimal surfaces in either a sphere $\Sf^4$ or a hyperbolic space $\Hy^4$ by an stereographic projection. We also determine the relation between the pairs $(g,h)$ of conjugate minimal surfaces associated to a superconformal surface and its image by an inversion. In particular, this yields a new transformation for minimal surfaces in $\R^4$.

math.DG↗

arXiv2007

Polar actions on compact Euclidean hypersurfaces

Given an isometric immersion $f\colon M^n\to \R^{n+1}$ of a compact Riemannian manifold of dimension $n\geq 3$ into Euclidean space of dimension $n+1$, we prove that the identity component $Iso^0(M^n)$ of the isometry group $Iso(M^n)$ of $M^n$ admits an orthogonal representation $Φ\colon Iso^0(M^n)\to SO(n+1)$ such that $f\circ g=Φ(g)\circ f$ for every $g\in Iso^0(M^n)$. If $G$ is a closed connected subgroup of $Iso(M^n)$ acting locally polarly on $M^n$, we prove that $Φ(G)$ acts polarly on $\R^{n+1}$, and we obtain that $f(M^n)$ is given as $Φ(G)(L)$, where $L$ is a hypersurface of a section which is invariant under the Weyl group of the $Φ(G)$-action. We also find several sufficient conditions for such an $f$ to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension $n\geq 3$ are characterized by their underlying warped product structure.

math.DG↗

arXiv2007

Isothermic submanifolds of Euclidean space

We study the problem posed by F. Burstall of developing a theory of isothermic Euclidean submanifolds of dimension greater than or equal to three. As a natural extension of the definition in the surface case, we call a Euclidean submanifold {\it isothermic} if it is locally the image of a conformal immersion of a Riemannian product of Riemannian manifolds whose second fundamental form is adapted to the product net of the manifold. Our main result is a complete classification of all such conformal immersions of Riemannian products of dimension greater than or equal to three. We derive several consequences of this result. We also study whether the classical characterizations of isothermic surfaces as solutions of Christoffel's problem and as envelopes of nontrivial conformal sphere congruences extend to higher dimensions.

math.DG↗

arXiv2006

Blaschke's problem for hypersurfaces

We solve Blaschke's problem for hypersurfaces of dimension $n\geq 3$. Namely, we determine all pairs of Euclidean hypersurfaces $f,\tilde{f}\colon M^n\to\R^{n+1}$ that induce conformal metrics on $M^n$ and envelope a common sphere congruence in $\R^{n+1}$.

math.DG↗

arXiv2004

Conformal de Rham decomposition of Riemannian manifolds

We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We also obtain other related de Rham-type decomposition theorems. As an application, we study Riemannian manifolds that admit a Codazzi tensor with two distinct eigenvalues everywhere.

math.DG↗

arXiv2004

Reducibility of Dupin submanifolds

We introduce the notion of weak reduciblity for Dupin submanifolds with arbitrary codimension. We give a complete characterization of all weakly reducible Dupin submanifolds, as a consequence of a general result on a broader class of Euclidean submanifolds. As a main application, we derive an explicit recursive procedure to generate all holonomic Dupin submanifolds in terms of solutions of completely integrable systems of linear partial differential equations of first order. We obtain several additional results on Dupin submanifolds.

math.DG↗

arXiv2002

Copolarity of isometric actions

We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity k representations, we classify the irreducible representations of copolarity one, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Samelson.

math.DG↗