arXiv subjects
Publications and source records attributed to Ruy Tojeiro.
We investigate the deformable hypersurfaces of the conformally flat Riemannian products $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$, $2\leq k\leq n-1$, of hyperbolic space and a sphere with constant sectional curvatures $-1$ and $1$, respectively.
We define hypersurfaces $f\colon M^n\to \mathbb{Q}_{c_1}^{k} \times \mathbb{Q}_{c_2}^{n-k+1}$ in class $\mathcal{A}$ of a product of two space forms as those that have flat normal bundle when regarded as submanifolds of the underlying flat ambient space. We provide an explicit construction of all of them in terms of parallel families of hypersurfaces of the factors, and show how such construction simplifies for the hypersurfaces within this class that have constant product angle function. We also show that hypersurfaces with constant mean curvature in class $\mathcal{A}$ are given in terms of parallel families of isoparametric hypersurfaces in each factor and a solution of a second order ODE. Finally, we classify hypersurfaces with constant mean curvature in class~$\mathcal{A}$ that have constant product angle function.
We classify the hypersurfaces of dimension n >= 3 with constant sectional curvature in the product spaces R^k x S^{n-k+1} and R^k x H^{n-k+1}, for 2 <= k <= n-1. Our results provide a complete description of these hypersurfaces and extend previous classifications of constant curvature submanifolds in product spaces of space forms.
In this article we classify the conformally flat Euclidean hypersurfaces of dimension three with three distinct principal curvatures of $\mathbb{R}^4$, $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$ with the property that the tangent component of the vector field $\partial/\partial t$ is a principal direction at any point. Here $\partial/\partial t$ stands for either a constant unit vector field in $\mathbb{R}^4$ or the unit vector field tangent to the factor $\mathbb{R}$ in the product spaces $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$, respectively. Then we use this result to give a simple proof of an alternative classification of the cyclic conformally flat hypersurfaces of $\mathbb{R}^4$, that is, the conformally flat hypersurfaces of $\mathbb{R}^4$ with three distinct principal curvatures such that the curvature lines correspondent to one of its principal curvatures are extrinsic circles. We also characterize the cyclic conformally flat hypersurfaces of $\mathbb{R}^4$ as those conformally flat hypersurfaces of dimension three with three distinct principal curvatures for which there exists a conformal Killing vector field of $\mathbb{R}^4$ whose tangent component is an eigenvector field correspondent to one of its principal curvatures.
Given a Riemannian manifold $N^n$ and ${\cal Z}\in \mathfrak{X}(N)$, an isometric immersion $f\colon M^m\to N^n$ is said to have the \emph{constant ratio property with respect to ${\cal Z}$} either if the tangent component ${\cal Z}^T_f$ of ${\cal Z}$ vanishes identically or if ${\cal Z}^T_f$ vanishes nowhere and the ratio $\|{\cal Z}^\perp_f\|/\|{\cal Z}^T_f\|$ between the lengths of the normal and tangent components of ${\cal Z}$ is constant along $M^m$. It has the \emph{principal direction property with respect to ${\cal Z}$} if ${\cal Z}^T_f$ is an eigenvector of all shape operators of $f$ at all points of $M^m$. In this article we study isometric immersions $f\colon M^m\to N^n$ of arbitrary codimension that have either the constant ratio or the principal direction property with respect to distinguished vector fields ${\cal Z}$ on space forms, product spaces $\Sf^n\times \R$ and $\Hy^n\times \R$, where $\Sf^n$ and $\Hy^n$ are the $n$-dimensional sphere and hyperbolic space, respectively, and, more generally, on warped products $I\times_ρ\Q_\e^n$ of an open interval $I\subset \R$ and a space form $\Q_\e^n$. Starting from the observation that these properties are invariant under conformal changes of the ambient metric, we provide new characterization and classification results of isometric immersions that satisfy either of those properties, or both of them simultaneously, for several relevant instances of ${\cal Z}$ as well as simpler descriptions and proofs of some known ones for particular cases of ${\cal Z}$ previously considered by many authors.
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 $f\colon M^n \rightarrow \mathbb{R}^{n+1}$ 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 $\tilde f|_U=h\circ f|_U$ of $f|_U$ with a conformal immersion $h\colon V\to \mathbb{R}^{n+2}$ of an open subset $V\subset \mathbb{R}^{n+1}$ containing $f(U)$.
We study conformally flat hypersurfaces $f\colon M^{3} \to \Q^{4}(c)$ with three distinct principal curvatures and constant mean curvature $H$ in a space form with constant sectional curvature $c$. First we extend a theorem due to Defever when $c=0$ and show that there is no such hypersurface if $H\neq 0$. Our main results are for the minimal case $H=0$. If $c\neq 0$, we prove that if $f\colon M^{3} \to \Q^{4}(c)$ is a minimal conformally flat hypersurface with three distinct principal curvatures then $f(M^3)$ is an open subset of a generalized cone over a Clifford torus in an umbilical hypersurface $\Q^{3}(\tilde c)\subset \Q^4(c)$, $\tilde c>0$, with $\tilde c\geq c$ if $c>0$. For $c=0$, we show that, besides the cone over the Clifford torus in $\Sf^3\subset \R^4$, there exists precisely a one-parameter family of (congruence classes of) minimal isometric immersions $f\colon M^3 \to \R^4$ with three distinct principal curvatures of simply-connected conformally flat Riemannian manifolds.
We obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds of constant sectional curvature of space forms, which we call the $L$-transformation. It allows to construct a family of such submanifolds starting with a given one and a vector-valued solution of a system of linear partial differential equations. We prove a decomposition theorem for the $L$-transformation, which is a far-reaching generalization of the classical permutability formula for the Ribaucour transformation of surfaces of constant curvature in Euclidean three space. As a consequence, we derive a Bianchi-cube theorem, which allows to produce, from $k$ initial scalar $L$-transforms of a given submanifold of constant curvature, a whole $k$-dimensional cube all of whose remaining $2^k-(k+1)$ vertices are submanifolds with the same constant sectional curvature given by explicit algebraic formulae. We also obtain further reductions, as well as corresponding decomposition and Bianchi-cube theorems, for the classes of $n$-dimensional flat Lagrangian submanifolds of $\mathbb{C}^n$ and $n$-dimensional Lagrangian submanifolds with constant curvature $c$ of the complex projective space $\mathbb C\mathbb P^n(4c)$ or the complex hyperbolic space $\mathbb C\mathbb H^n(4c)$ of complex dimension $n$ and constant holomorphic curvature~4c.
We give a complete classification of umbilical surfaces of arbitrary codimension of a product $Q^{n_1}_{k_1}\times Q^{n_2}_{k_2}$ of space forms whose curvatures satisfy $k_1 + k_2 \not= 0$.
In this paper, we analyze the geometric structure of an Euclidean submanifold whose osculating spaces form a nonconstant family of proper subspaces of the same dimension. We prove that if the rate of change of the osculating spaces is small, then the submanifold must be a (submanifold of a) ruled submanifold of a very special type. We also give a sharp estimate of the dimension of the rulings.
We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold endowed with a polar metric. For a product manifold endowed with a polar metric, our main result gives a complete description of all its isometric immersions into a space form whose second fundamental forms are adapetd to its product structure, in the sense that the tangent spaces to each factor are preserved by all shape operators. This is a far-reaching generalization of a basic decomposition theorem for isometric immersions of Riemannian products due to Moore as well as its extension by Nölker to isometric immersions of warped products.
We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension $m\geq 3$ of a product $\Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2\neq 0$ to the classification of $m$-dimensional umbilical submanifolds of codimension two of $\Sf^n\times \R$ and $\Hy^n\times \R$. The case of $\Sf^n\times \R$ was carried out in \cite{mt}. As a main tool we derive reduction of codimension theorems of independent interest for submanifolds of products of two space forms.
We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of $\Sf^n\times \R$, extending the classification of umbilical surfaces in $\Sf^2\times \R$ by Rabah-Souam and Toubiana as well as the local description of umbilical hypersurfaces in $\Sf^n\times \R$ by Van der Veken and Vrancken. We prove that, besides small spheres in a slice, up to isometries of the ambient space they come in a two-parameter family of rotational submanifolds whose substantial codimension is either one or two and whose profile is a curve in a totally geodesic $\Sf^1\times \R$ or $\Sf^2\times \R$, respectively, the former case arising in a one-parameter family. All of them are diffeomorphic to a sphere, except for a single element that is diffeomorphic to Euclidean space. We obtain explicit parametrizations of all such submanifolds. We also study more general classes of submanifolds of $\Sf^n\times \R$ and $\Hy^n\times \R$. In particular, we give a complete description of all submanifolds in those product spaces for which the tangent component of a unit vector field spanning the factor $\R$ is an eigenvector of all shape operators. We show that surfaces with parallel mean curvature vector in $\Sf^n\times \R$ and $\Hy^n\times \R$ having this property are rotational surfaces. We also prove a Dajczer-type reduction of codimension theorem for submanifolds of $\Sf^n\times \R$ and $\Hy^n\times \R$.
We classify hypersurfaces of rank two of Euclidean space $\R^{n+1}$ that admit genuine isometric deformations in $\R^{n+2}$. That an isometric immersion $\hat f\colon\,M^n\to\R^{n+2}$ is a genuine isometric deformation of a hypersurface $f\colon\, M^n\to\R^{n+1}$ means that $\hat f$ is nowhere a composition $\hat f=\hat F\circ f$, where $\hat F\colon\,V\subset \R^{n+1}\to\R^{n+2}$ is an isometric immersion of an open subset $V$ containing $f(M)$.
We give a complete solution of a problem in submanifold theory posed and partially solved by the eminent algebraic geometer Pierre Samuel in 1947. Namely, to determine all pairs of immersions of a given manifold into Euclidean space that have the same Gauss map and induce conformal metrics on the manifold. The case of isometric induced metrics was solved in 1985 by the first author and D. Gromoll.
We classify the hypersurfaces of $\Sf^n\times \R$ and $\Hy^n\times \R$ with constant sectional curvature and dimension $n\geq 3$.
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.