Searcharxiv⌕ Search

arXiv subjects

Francisco Vittone

Publications and source records attributed to Francisco Vittone.

13 recordsLinked to original sources
arXiv2026

On the structure of $2$-step nilpotent Lorentzian naturally reductive Lie groups

We study $2$-step nilpotent Lie groups with naturally reductive left-invariant Lorentzian metrics with respect to the presentation group $N \rtimes H^{\operatorname{aut}}$. Replacing the standard non-degenerate center assumption with the weaker condition that the commutator ideal be non-degenerate, we develop a framework that extends the construction to the Lorentzian context and covers both the non-degenerate and degenerate center cases. In the degenerate case, we show that the associated Lie algebra is a central extension of a semidirect product whose Riemannian factor is naturally reductive. Furthermore, we obtain invariant decompositions of the defining representation, including a distinguished Lorentzian factor, and provide an explicit description of the isotropy algebra and the identity component of the isometric automorphism group. These results complete the structural description of naturally reductive $2$-step Lorentzian nilpotent Lie groups under the assumption of non-degeneracy in the commutator ideal.

math.DG↗
arXiv2025

The distribution of symmetry of Lorentzian naturally reductive nilmanifolds

We study $2$-step nilpotent Lorentzian Lie groups $N$, which are naturally reductive with respect to a certain class of transitive subgroups of isometries. We describe the isotropy representation and prove that its fixed points give raise to the distribution of symmetry of $N$. This generalizes some known results for the Riemannian case.

math.DG↗
arXiv2022

The structure of homogeneous Riemannian manifolds with nullity

We find new conditions that the existence of nullity of the curvature tensor of an irreducible homogeneous space $M=G/H$ imposes on the Lie algebra $\mathfrak g$ of $G$ and on the Lie algebra $\tilde{\mathfrak g}$ of the full isometry group of $M$. Namely, we prove that there exists a transvection of $M$ in the direction of any element of the nullity, possibly by enlarging the presentation group $G$. Moreover, we prove that these transvections generate an abelian ideal of $\tilde{\mathfrak g}$. These results constitute a substantial improvement on the structure theory developed in \cite{DOV}. In addition we construct examples of homogeneous Riemannian spaces with non-trivial nullity, where $G$ is a non-solvable group, answering a natural open question. Such examples admit (locally homogeneous) compact quotients. In the case of co-nullity $3$ we give an explicit description of the isometry group of any homogeneouslocally irreducible Riemannian manifold with nullity.

math.DG↗
arXiv2022

Mobius fluid dynamics on the unitary groups]{Möbius fluid dynamics on the unitary groups

We study the nonrigid dynamics induced by the standard birational actions of the split unitary groups $G=O_{o}\left( n,n\right) $, $SU\left( n,n\right) $ and $Sp\left( n,n\right) $ on the compact classical Lie groups $M=SO_{n}$, $% U_{n}$ and $Sp_{n}$, respectively. More precisely, we study the geometry of $% G$ endowed with the kinetic energy metric associated with the action of $G$ on $M,$ assuming that $M$ carries its canonical bi-invariant Riemannian metric and has initially a homogeneous distribution of mass. By the least action principle, force free motions (thought of as curves in $G$) correspond to geodesics of $G$. The geodesic equation may be understood as an inviscid Burgers equation with Möbius constraints. We prove that the kinetic energy metric on $G$ is not complete and in particular not invariant, find symmetries and totally geodesic submanifolds of $G$ and address the question under which conditions geodesics of rigid motions are geodesics of $G$. Besides, we study equivalences with the dynamics of conformal and projective motions of the sphere in low dimensions.

math.DG↗
arXiv2020

Cones and Cartan geometry

We show that the extended principal bundle of a Cartan geometry of type $(A(m,\mathbb{R}),GL(m,\mathbb{R}))$, endowed with its extended connection $\hatω$, is isomorphic to the principal $A(m,\mathbb{R})$-bundle of affine frames endowed with the affine connection as defined in classical Kobayashi-Nomizu volume I. Then we classify the local holonomy groups of the Cartan geometry canonically associated to a Riemannian manifold. It follows that if the holonomy group of the Cartan geometry canonically associated to a Riemannian manifold is compact then the Riemannian manifold is locally a product of cones.

math.DG↗
arXiv2020

Homogeneous Riemannian manifolds with non-trivial nullity

We develop a general theory for irreducible homogeneous spaces $M= G/H$, in relation to the nullity $ν$ of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e. where the nullity distribution is not parallel. Moreover, we construct irreducible examples of conullity $k=3$, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that $H$ is trivial and $G$ is solvable if $k=3$. Another of our main results is that the leaves of the nullity are closed (we used a rather delicate argument). This implies that $M$ is a Euclidean affine bundle over the quotient by the leaves of $ν$. Moreover, we prove that $ν^\perp$ defines a metric connection on this bundle with transitive holonomy or, equivalently, $ν^\perp$ is completely non-integrable (this is not in general true for an arbitrary autoparallel and flat invariant distribution). We also found some general obstruction for the existence of non-trivial nullity: e.g., if $G$ is reductive (in particular, if $M$ is compact), or if $G$ is two-step nilpotent.

math.DG↗
arXiv2017

Mok's characteristic varieties and the normal holonomy group

In this paper we complete the study of the normal holonomy groups of complex submanifolds (non nec. complete) of Cn or CPn. We show that irreducible but non transitive normal holonomies are exactly the Hermitian s-representations of [CD09, Table 1] (see Corollary 1.1). For each one of them we construct a non necessarily complete complex submanifold whose normal holonomy is the prescribed s-representation. We also show that if the submanifold has irreducible non transitive normal holonomy then it is an open subset of the smooth part of one of the characteristic varieties studied by N. Mok in his work about rigidity of locally symmetric spaces. Finally, we prove that if the action of the normal holonomy group of a projective submanifold is reducible then the submanifold is an open subset of the smooth part of a so called join, i.e. the union of the lines joining two projective submanifolds.

math.DG↗
arXiv2015

Normal holonomy of CR submanifolds

We study the normal holonomy group, i.e. the holonomy group of the normal connection, of a CR-submanifold of a complex space form. We complete the local classification of normal holonomies for complex submanifolds. We show that the normal holonomy group of a coisotropic submanifold acts as the holonomy representation of a Riemannian symmetric space. In case of a totally real submanifold we give two results about reduction of codimension. We describe explicitly the action of the normal holonomy in the case in which the totally real submanifold is contained in a totally real totally geodesic submanifold. In such a case we prove the compactness of the normal holonomy group.

math.DG↗
arXiv2013

Codimension reduction in symmetric spaces

In this paper we give a short geometric proof of a generalization of a well-known result about reduction of codimension for submanifolds of Riemannian symmetric spaces.

math.DG↗
arXiv2012

Naturally reductive pseudo-Riemannian Lie groups in low dimensions

This work concerns the non-flat metrics on the Heisenberg Lie group of dimension three $\Heis_3(\RR)$ and the bi-invariant metrics on the solvable Lie groups of dimension four. On $\Heis_3(\RR)$ we prove that the property of the metric being naturally reductive is equivalent to the property of the center being non-degenerate. These metrics are Lorentzian algebraic Ricci solitons. We start with the indecomposable Lie groups of dimension four admitting bi-invariant metrics and which act on $\Heis_3(\RR)$ by isometries and we finally study some geometrical features on these spaces.

math.DG↗
arXiv2011

On the nullity distribution of the second fundamental form of a submanifold of a space form

If M is a submanifold of a space form, the nullity distribution N of its second fundamental form is (when defined) the common kernel of its shape operators. In this paper we will give a local description of any submanifold of the Euclidean space by means of its nullity distribution. We will also show the following global result: if M is a complete, irreducible submanifold of the Euclidean space or the sphere then N is completely non integrable. This means that any two points in M can be joined by a curve everywhere perpendicular to N. We will finally show that this statement is false for a submanifold of the hyperbolic space.

math.DG↗