arXiv subjects

Luis A. Florit

Publications and source records attributed to Luis A. Florit.

12 recordsLinked to original sources

arXiv2022

Scalar positive immersions

As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive scalar curvature metrics on closed simply connected manifolds in dimensions at least five appears on spin manifolds and is given by the non-vanishing of the $α$-genus of Hitchin. When unobstructed we shall realize a positive scalar curvature metric by an immersion into Euclidean space whose dimension is uniformly close to the classical Whitney upper bound for smooth immersions. Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure for constructing positive scalar curvature metrics.

math.DG↗

arXiv2018

Singular genuine rigidity

We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${\mathbb R}^{n+p}$ is singularly genuinely rigid in ${\mathbb R}^{n+q}$, for any $q < \min\{5,n\} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are removed.

math.DG↗

arXiv2017

Manifolds with conullity at most two as graph manifolds

We find necessary and sufficient conditions for a complete $n$-dimensional Riemannian manifold of finite volume, whose curvature tensor has nullity at least $n-2$, to be a geometric graph manifold. In the process, we show that Nomizu's conjecture, well known to be false in general, is true for manifolds with finite volume.

math.DG↗

arXiv2016

Nonnegatively curved Euclidean submanifolds in codimension two

We provide a classification of compact Euclidean submanifolds $M^n\subset{\mathbb{R}}^{n+2}$ with nonnegative sectional curvature, for $n\ge 3$. The classification is in terms of the induced metric (including the diffeomorphism classification of the manifold), and we study the structure of the immersions as well. In particular, we provide the first known example of a nonorientable quotient $({\mathbb{S}}^{n-1}\times{\mathbb{S}}^1)/{\mathbb{Z}_2}\subset{\mathbb{R}}^{n+2}$ with nonnegative curvature. For the 3-dimensional case, we show that either the universal cover is isometric to ${\mathbb{S}}^2\times{\mathbb{R}}$, or $M^3$ is diffeomorphic to a lens space, and the complement of the (nonempty) set of flat points is isometric to a twisted cylinder $(N^2\times{\mathbb{R}})/{\mathbb{Z}}$. As a consequence we conclude that, if the set of flat points is not too big, there exists a unique flat totally geodesic surface in $M^3$ whose complement is the union of one or two twisted cylinders over disks.

math.DG↗

arXiv2016

Classification of codimension two deformations of rank two Riemannian manifolds

The purpose of this work is to close the local deformation problem of rank two Euclidean submanifolds in codimension two by describing their moduli space of deformations. In the process, we provide an explicit simple representation of these submanifolds, a result of independent interest by its applications. We also determine which deformations are genuine and honest, allowing us to find the first known examples of honestly locally deformable rank two submanifolds in codimension two. In addition, we study which of these submanifolds admit isometric immersions as Euclidean hypersurfaces, a property that gives rise to several applications to the Sbrana-Cartan theory of deformable Euclidean hypersurfaces.

math.DG↗

arXiv2011

Topological obstructions to fatness

Alan Weinstein showed that certain characteristic numbers of any Riemannian submersion with totally geodesic fibers and positive vertizontal curvatures are nonzero. In this paper we explicitly compute these invariants in terms of Chern and Pontrjagin numbers of the bundle. This allows us to show that many bundles do not admit such metrics.

math.DG↗

arXiv2008

The holomorphic Gauss Parametrization

We give a local parametric description of all holomorphic hypersurfaces in complex Euclidean and projective spaces with constant index of relative nullity, together with applications. This is a complex analogue to the parametrization for real hypersurfaces in Euclidean space known as the Gauss parametrization.

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↗

arXiv2006

Complete real Kahler submanifolds in codimension two

Minimal isometric immersions F in codimension two from a complete Kahler manifold into Euclidean space had been classified for dimension greater than or equal to 3. In this note we describe the non--minimal situation by showing that, if F is real analytic but not everywhere minimal, then F is a cylinder over a real Kahler surface G in Euclidean 6-space. In addition, G can be further described.

math.DG↗

arXiv2006

On the topology of positively curved Bazaikin spaces

We study the topology of the 13 dimensional positively curved Bazaikin spaces. We show that there is only one such manifold which is homotopy equivalent to a homogeneous space, the so called Berger space. This is in contrast to the case of the 7 dimensional positively curved Eschenburg spaces. In addition, we compute the Pontryagin classes and the linking form and show that the first two billion positively curved Bazaikin manifolds are homeomorphically distinct, raising the question whether this is true in general.

math.DG↗

arXiv2006

On Yamabe constants of Riemannian products

For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as $r$ goes to infinity is equal to the Yamabe constant of $(M^m \times R^n, [g+g_E])$ and is strictly less than the Yamabe invariant of $S^{m+n}$ provided $n\geq 2$. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the Gagliardo-Nirenberg inequalities.

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↗