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Roberto Rubio

Publications and source records attributed to Roberto Rubio.

At least 19 recordsLinked to original sources
arXiv2026

On the Dirac complement problem

The existence of a Dirac complement for a given Dirac structure is a central question in the structure theory of Courant algebroids and the deformation theory of Dirac structures. We study this problem in detail, proving the unobstructedness of lagrangian or local Dirac complements and providing examples that show the complexity of this question. We introduce a cohomology class whose nonvanishing prevents the existence of a Dirac complement and apply it to several families of examples. On the other hand, by using Lie-theoretical techniques, we prove that, for a Lie algebra $\mathfrak{g}$ endowed with a definite form, the diagonal in $\mathfrak{g} \oplus \bar{\mathfrak{g}}$ does not admit a complement unless $\mathfrak{g}$ is abelian. This includes real compact semisimple Lie algebras with their Killing form.

math.SG↗
arXiv2025

Courant algebroid lifts and curved Courant algebroids

We introduce the Courant algebroid lift, a new construction that takes a Courant algebroid together with a vector bundle connection and produces, when the connection is flat in the image of the anchor, a Courant algebroid. In general, this lift produces a Courant-like structure that we call a curved Courant algebroid. We start by establishing a hierarchy of Courant algebroid properties and their associated structures. In this setting, we introduce curved Courant algebroids, which we show to be related to connections with torsion and curved differential graded Lie algebras. We use this to provide a classification of exact curved Courant algebroids. We show that the Courant algebroid lift of an exact Courant algebroid yields a natural link between the Patterson-Walker metric and generalized geometry. By lifting non-exact Courant algebroids, we establish a relation of these lifts to Lie algebras, Poisson and special complex geometry. Finally, we show that Courant algebroid lifts provide a large class of examples of Courant algebroid actions.

math.DG↗
arXiv2025

Symmetric Poisson geometry, totally geodesic foliations and Jacobi-Jordan algebras

We introduce symmetric Poisson structures as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying an integrability condition analogous to that in usual Poisson geometry. Equivalent conditions in Poisson geometry have inequivalent analogues in symmetric Poisson geometry and we distinguish between symmetric and strong symmetric Poisson structures. We prove that symmetric Poisson structures correspond to locally geodesically invariant distributions together with a characteristic metric, whereas strong symmetric Poisson structures correspond to totally geodesic foliations together with a leaf metric and a leaf connection. We introduce, using the Patterson-Walker metric, a dynamics on the cotangent bundle and show its connection to symmetric Poisson geometry, the parallel transport equation and the Newtonian equation for conservative systems. Finally, we prove that linear symmetric Poisson structures are in correspondence with Jacobi-Jordan algebras, whereas strong symmetric correspond to those that are moreover associative.

math.DG↗
arXiv2025

Flat Generalized Connections on Courant Algebroids

We consider a family of metric generalized connections on transitive Courant algebroids, which includes the canonical Levi-Civita connection, and study the flatness condition. We find that the building blocks for such flat transitive Courant algebroids are compact simple Lie groups. Further, we give a description of left-invariant flat Levi-Civita generalized connections on such Lie groups, which, in particular, shows the existence of non-flat ones.

math.DG↗
arXiv2025

Symmetric Cartan calculus, the Patterson-Walker metric and Killing vector fields

We develop symmetric Cartan calculus, an analogue of classical Cartan calculus for symmetric differential forms. We first show that the analogue of the exterior derivative, the symmetric derivative, is not unique and its different choices are parametrized by torsion-free affine connections. We use a choice of symmetric derivative to generate the symmetric Lie derivative and the symmetric bracket, and give geometric interpretations of all of them. By proving the structural identities and describing the role of affine morphisms, we reveal an unexpected link of symmetric Cartan calculus with the Patterson-Walker metric, which we recast as a direct analogue of the canonical symplectic form on the cotangent bundle. We show that, in the light of the Patterson-Walker metric, symmetric Cartan calculus becomes a complete analogue of classical Cartan calculus. In this analogy, its Killing vector fields play a central role.

math.DG↗
arXiv2024

On the Equivalence of Generalized Ricci Curvatures

We prove the equivalence between the several notions of generalized Ricci curvature found in the literature. As an application, we characterize when the total generalized Ricci tensor is symmetric.

math.DG↗
arXiv2024

New geometric structures on 3-manifolds: surgery and generalized geometry

Cosymplectic and normal almost contact structures are analogues of symplectic and complex structures that can be defined on 3-manifolds. Their existence imposes strong topological constraints. Generalized geometry offers a natural common generalization of these two structures: $B_3$-generalized complex structures. We prove that any closed orientable 3-manifold admits such a structure, which can be chosen to be stable, that is, generically cosymplectic up to generalized diffeomorphism.

math.DG↗
arXiv2021

On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievability

We compute the contact manifold of null geodesics of the family of spacetimes $\{(\mathbb{S}^2\times\mathbb{S}^1, g_\circ-\frac{d^2}{c^2}dt^2)\}_{d,c\in\mathbb{N}^+\text{ coprime}}$, with $g_\circ$ the round metric on $\mathbb{S}^2$ and $t$ the $\mathbb{S}^1$-coordinate. We find that these are the lens spaces $L(2c,1)$ together with the pushforward of the canonical contact structure on $ST\mathbb{S}^2\cong L(2,1)$ under the natural projection $L(2,1)\to L(2c,1)$. We extend this computation to $Z\times \mathbb{S}^1$ for $Z$ a Zoll manifold. On the other hand, motivated by these examples, we show how Engel geometry can be used to describe the manifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation. We characterize the three-dimensional contact manifolds that are contactomorphic to the space of null geodesics of a spacetime. The characterization consists in the existence of an overlying Engel manifold with a certain foliation and, in this case, we also retrieve the spacetime.

math.DG↗
arXiv2021

Complex Dirac structures: invariants and local structure

We study complex Dirac structures, that is, Dirac structures in the complexified generalized tangent bundle. These include presymplectic foliations, transverse holomorphic structures, CR-related geometries and generalized complex structures. We introduce two invariants, the order and the (normalized) type. We show that, together with the real index, they allow us to obtain a pointwise classification of complex Dirac structures. For constant order, we prove the existence of an underlying real Dirac structure, which generalizes the Poisson structure associated to a generalized complex structure. For constant real index and order, we prove a splitting theorem, which gives a local description in terms of a presymplectic leaf and a small transversal.

math.DG↗
arXiv2020

Gauge theory for string algebroids

We introduce a moment map picture for holomorphic string algebroids where the Hamiltonian gauge action is described by means of inner automorphisms of Courant algebroids. The zero locus of our moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull-Strominger system. Our main results are concerned with the geometry of the moduli space of solutions, and assume a technical condition which is fulfilled in examples. We prove that the moduli space carries a pseudo-K\"ahler metric with K\"ahler potential given by the dilaton functional, a topological formula for the metric, and an infinitesimal Donaldson-Uhlenbeck-Yau type theorem.

math.DG↗
arXiv2019

On the Gelfand property for complex symmetric pairs

We first prove, for pairs consisting of a simply connected complex reductive group together with a connected subgroup, the equivalence between two different notions of Gelfand pairs. This partially answers a question posed by Gross, and allows us to use a criterion due to Aizenbud and Gourevitch, and based on Gelfand-Kazhdan's theorem, to study the Gelfand property for complex symmetric pairs. This criterion relies on the regularity of the pair and its descendants. We introduce the concept of a pleasant pair, as a means to prove regularity, and study, by recalling the classification theorem, the pleasantness of all complex symmetric pairs. On the other hand, we prove a method to compute all the descendants of a complex symmetric pair by using the extended Satake diagram, which we apply to all pairs. Finally, as an application, we prove that eight out of the twelve exceptional complex symmetric pairs, together with the infinite family $(\textrm{Spin}_{4q+2}, \textrm{Spin}_{4q+1})$, satisfy the Gelfand property, and state, in terms of the regularity of certain symmetric pairs, a sufficient condition for a conjecture by van Dijk and a reduction of a conjecture by Aizenbud and Gourevitch.

math.RT↗
arXiv2018

Holomorphic string algebroids

We introduce the category of holomorphic string algebroids, whose objects are Courant extensions of Atiyah Lie algebroids of holomorphic principal bundles, as considered by Bressler, and whose morphisms correspond to inner morphisms of the underlying holomorphic Courant algebroids in the sense of Severa. This category provides natural candidates for Atiyah Lie algebroids of holomorphic principal bundles for the (complexified) string group and their morphisms. Our main results are a classification of string algebroids in terms of Cech cohomology, and the construction of a locally complete family of deformations of string algebroids via a differential graded Lie algebra.

math.AG↗
arXiv2018

Canonical metrics on holomorphic Courant algebroids

The solution of the Calabi Conjecture by Yau implies that every K\"ahler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\textrm{SU}(n)$, and that these metrics are parametrized by the positive cone in $H^{1,1}(X,\mathbb{R})$. In this work we give evidence of an extension of Yau's theorem to non-K\"ahler manifolds, where $X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid $Q$ of Bott-Chern type. The equations that define our notion of best metric correspond to a mild generalization of the Hull-Strominger system, whereas the role of $H^{1,1}(X,\mathbb{R})$ is played by an affine space of 'Aeppli classes' naturally associated to $Q$ via Bott-Chern secondary characteristic classes.

math.DG↗
arXiv2016

The Lie group of automorphisms of a Courant algebroid and the moduli space of generalized metrics

We endow the group of automorphisms of an exact Courant algebroid over a compact manifold with an infinite dimensional Lie group structure modelled on the inverse limit of Hilbert spaces (ILH). We prove a slice theorem for the action of this Lie group on the space of generalized metrics. As an application, we show that the moduli space of generalized metrics is stratified by ILH submanifolds and relate it to the moduli space of usual metrics. Finally, we extend these results to odd exact Courant algebroids.

math.DG↗
arXiv2016

On higher Dirac structures

We study higher-order analogues of Dirac structures, extending the multisymplectic structures that arise in field theory. We define higher Dirac structures as involutive subbundles of $TM+\wedge^k TM^*$ satisfying a weak version of the usual lagrangian condition (which agrees with it only when $k=1$). Higher Dirac structures transversal to $TM$ recover the higher Poisson structures introduced in [8] as the infinitesimal counterparts of multisymplectic groupoids. We describe the leafwise geometry underlying an involutive isotropic subbundle in terms of a distinguished 1-cocycle in a natural differential complex, generalizing the presymplectic foliation of a Dirac structure. We also identify the global objects integrating higher Dirac structures.

math.SG↗
arXiv2015

Higgs bundles, the Toledo invariant and the Cayley correspondence

We define the Toledo invariant of a G-Higgs bundle on a Riemann surface, where G is a real semisimple group of Hermitian type, and we prove a Milnor-Wood type bound for this invariant when the bundle is semistable. We prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley correspondence when the symmetric space defined by G is of tube type. This gives a new proof of the Milnor-Wood inequality of Burger-Iozzi-Wienhard for representations of the fundamental group of a Riemann surface into G. Compared to previous results using Higgs bundles, it uses general theory and avoids any case by case study.

math.DG↗
arXiv2015

Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry

We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable Killing spinor equations on a Courant algebroid. As an application, we propose a unifying framework for metrics with holonomy $\SU(3)$ and solutions of the Strominger system.

math.DG↗