Preprints

14. The submanifold compatibility equations in magnetic geometry
    Submitted for publication.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    With the notions of magnetic curvature and magnetic second fundamental form recently introduced by Assenza and Albers-Benedetti-Maier, respectively, we establish analogues of the Gauss, Ricci, and Codazzi–Mainardi compatibility equations from submanifold theory in the magnetic setting.
15. Parallel differential forms of codegree two, and three-forms in dimension six (with A. Derdzinski and P. Piccione)
    Submitted for publication.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    For a differential form on a manifold, having constant components in suitable local coordinates trivially implies being parallel relative to a torsion-free connection, and the converse implication is known to be true for $p$-forms in dimension $n$ when $p = 0, 1, 2, n- 1, n$. We prove the converse for $(n - 2)$-forms, and for $3$-forms when $n = 6$, while pointing out that it fails to hold for Cartan $3$-forms on all simple Lie groups of dimensions $n \geq 8$ as well as for $(n, p) = (7, 3)$ and $(n, p) = (8, 4)$, where the $3$-forms and $4$-forms arise in compact simply connected Riemannian manifolds with exceptional holonomy groups. We also provide geometric characterizations of $3$-forms in dimension six and $(n-2)$-forms in dimension n having the constant-components property mentioned above, and describe examples illustrating the fact that various parts of these geometric characterizations are logically independent.
16. Marked length spectrum rigidity for Anosov magnetic surfaces (with V. Assenza, J. de Simoi, and J. Marshall Reber)
    Submitted for publication.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    We show that if $M$ is a closed, connected, oriented surface, and two Anosov magnetic systems on $M$ are conjugate by a volume-preserving conjugacy isotopic to the identity, with their magnetic forms in the same cohomology class, then the metrics are isometric. This extends the recent result of Guillarmou, Lefeuvre, and Paternain to the magnetic setting.

Publications

11. Nijenhuis geometry of parallel tensors (with A. Derdzinski and P. Piccione)
    Annali di Matematica Pura ed Applicata, vol. 204 (2025), no. 4, pp. 1381-1401.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    A tensor -- meaning here a tensor field $\Theta$ of any type $(p, q)$ on a manifold -- may be called integrable if it is parallel relative to some torsion-free connection. We provide analytical and geometric characterizations of integrability for differential $q$-forms, $q = 0, 1, 2, n - 1, n$ (in dimension $n$), vectors, bivectors, symmetric $(2, 0)$ and $(0, 2)$ tensors, as well as complex-diagonalizable and nilpotent tensors of type $(1, 1)$. In most cases, integrability is equivalent to algebraic constancy of $\Theta$ coupled with the vanishing of one or more suitably defined Nijenhuis-type tensors, depending on $\Theta$ via a quasilinear first-order differential operator. For $(p, q) = (1, 1)$, they include the ordinary Nijenhuis tensor.
12. Compact plane waves with parallel Weyl curvature
    To appear in Proceedings of the XI International Meeting in Lorentzian Geometry.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    This is an exposition of recent results -- obtained in joint work with Andrzej Derdzinski -- on essentially conformally symmetric (ECS) manifolds, that is, those pseudo-Riemannian manifolds with parallel Weyl curvature which are not locally symmetric or conformally flat. In the 1970s, Roter proved that while Riemannian ECS manifolds do not exist, pseudo-Riemannian ones do exist in all dimensions $n\geq 4$, and realize all indefinite metric signatures. The local structure of ECS manifolds is known, and every ECS manifold carries a distinguished null parallel distribution $\mathcal{D}$, whose rank is always equal to $1$ or $2$. We review basic facts about ECS manifolds, briefly discuss the construction of compact examples, and outline the proof of a topological structure result: outside of the locally homogeneous case and up to a double covering, every compact rank-one ECS manifold is a bundle over $\mathbb{S}^1$ whose fibers are the leaves of $\mathcal{D}^\perp$. Finally, we mention some classification results for compact rank-one ECS manifolds.
13. Magnetic flatness and E. Hopf's theorem for magnetic systems (with V. Assenza and J. Marshall Reber)
    Communications in Mathematical Physics, vol. 406 (2025), no. 2, art. 24.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    Using the notion of magnetic curvature recently introduced by the first author, we extend E. Hopf’s theorem to the setting of magnetic systems. Namely, we prove that if the magnetic flow on the $s$-sphere bundle is without conjugate points, then the total magnetic curvature is non-positive, and vanishes if and only if the magnetic system is magnetically flat. We then prove that magnetic flatness is a rigid condition, in the sense that it only occurs when either the magnetic form is trivial and the metric is flat, or when the magnetic system is Kähler, the metric has constant negative sectional holomorphic curvature, and $s$ equals the Mañé critical value.
14. Killing fields on compact pseudo-Kähler manifolds (with A. Derdzinski)
    Journal of Geometric Analysis, vol. 34 (2024), no. 5, art. 144.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    We show that a Killing field on a compact pseudo-Kähler $\partial\bar{\partial}$-manifold is necessarily (real) holomorphic. Our argument works without the $\partial\bar{\partial}$ assumption in real dimension four. The claim about holomorphicity of Killing fields on compact pseudo-Kähler manifolds appears in a 2012 paper by Yamada, and in an appendix we provide a detailed explanation of why we believe that Yamada’s argument is incomplete.
15. Codazzi tensor fields in reductive homogeneous spaces (with J. Marshall Reber)
    Results in Mathematics - Resultate der Mathematik, vol. 79 (2024), no. 4, art. 137.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    We extend the results about left-invariant Codazzi tensor fields on Lie groups equipped with left-invariant Riemannian metrics obtained by d’Atri in 1985 to the setting of reductive homogeneous spaces $G/H$, where the curvature of the canonical connection of second kind associated with the fixed reductive decomposition $\mathfrak{g} = \mathfrak{h}\oplus \mathfrak{m}$ enters the picture. In particular, we show that invariant Codazzi tensor fields on a naturally reductive homogeneous space are parallel.
16. Compact locally homogeneous manifolds with parallel Weyl tensor (with A. Derdzinski)
    Advances in Geometry, vol. 24 (2024), no. 4, pp. 493-503.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    We construct new examples of compact ECS manifolds, that is, of pseudo-Riemannian manifolds with parallel Weyl tensor that are neither conformally flat nor locally symmetric. Every ECS manifold has rank $1$ or $2$, the rank being the dimension of a distinguished null parallel distribution discovered by Olszak. Previously known examples of compact ECS manifolds, in every dimension greater than $4$, were all of rank $1$, geodesically complete, and none of them locally homogeneous. By contrast, our new examples -- all of them geodesically incomplete -- realize all odd dimensions starting from $5$ and are this time of rank $2$, as well as locally homogeneous.
17. The metric structure of compact rank-one ECS manifolds (with A. Derdzinski)
    Annals of Global Analysis and Geometry, vol. 64 (2023), no. 4, art. 24.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    Pseudo-Riemannian manifolds with nonzero parallel Weyl tensor which are not locally symmetric are known as ECS manifolds. Every ECS manifold carries a distinguished null parallel distribution $\mathcal{D}$, the rank $d\in \{1,2\}$ of which is referred to as the rank of the manifold itself. Under a natural genericity assumption on the Weyl tensor, we fully describe the universal coverings of compact rank-one ECS manifolds. We then show that any generic compact rank-one ECS manifold must be translational, in the sense that the holonomy group of the natural flat connection induced on $\mathcal{D}$ is either trivial or isomorphic to $\mathbb{Z}_2$. We also prove that all four-dimensional rank-one ECS manifolds are noncompact, this time without assuming genericity, as it is always the case in dimension four.
18. Rank-one ECS manifolds of dilational type (with A. Derdzinski)
    Portugaliae Mathematica, vol. 81 (2024), no. 1-2, pp. 69-96.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    We study ECS manifolds, that is, pseudo-Riemannian manifolds with parallel Weyl tensor which are neither conformally flat nor locally symmetric. Every ECS manifold has rank $1$ or $2$, the rank being the dimension of a distinguished null parallel distribution discovered by Olszak, and a rank-one ECS manifold may be called translational or dilational, depending on whether the holonomy group of a natural flat connection in the Olszak distribution is finite or infinite. Some such manifolds are in a natural sense generic, which refers to the algebraic structure of the Weyl tensor. Various examples of compact rank-one ECS manifolds are known: translational ones (both generic and nongeneric) in every dimension $n\geq 5$, as well as odd-dimensional nongeneric dilational ones, some of which are locally homogeneous. As we show, generic compact rank-one ECS manifolds must be translational or locally homogeneous, provided that they arise as isometric quotients of a specific class of explicitly constructed "model" manifolds. This result is relevant since the clause starting with "provided that" may be dropped: according to a theorem which we prove in another paper, the models just mentioned include the isometry types of the pseudo-Riemannian universal coverings of all generic compact rank-one ECS manifolds. Consequently, all generic compact rank-one ECS manifolds are translational.
19. Conformal flatness of compact three-dimensional Cotton-parallel manifolds
    Proceedings of the American Mathematical Society, vol. 152 (2024), no. 2, pp. 797-800.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    A three-dimensional pseudo-Riemannian manifold is called essentially conformally symmetric (ECS) if its Cotton tensor is parallel but nowhere-vanishing. In this note we prove that three-dimensional ECS manifolds must be noncompact or, equivalently, that every compact three-dimensional Cotton-parallel pseudo-Riemannian manifold must be conformally flat.
20. The topology of compact rank-one ECS manifolds (with A. Derdzinski)
    Proceedings of the Edinburgh Mathematical Society, vol. 66 (2023), no. 3, pp. 789-809.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    Pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as ECS manifolds, have a natural local invariant, the rank, which equals $1$ or $2$, and is the rank of a certain distinguished null parallel distribution $\mathcal{D}$. All known examples of compact ECS manifolds are of rank one and have dimensions greater than $4$. We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced when necessary by a two-fold isometric covering, must be a bundle over the circle with leaves of $\mathcal{D}^\perp$ serving as the fibres. The same conclusion holds in the locally-homogeneous case if one assumes that $\mathcal{D}^\perp$ has at least one compact leaf. We also show that in the pseudo-Riemannian universal covering space of any compact rank-one ECS manifold the leaves of $\mathcal{D}^\perp$ are the factor manifolds of a global product decomposition.
21. New examples of compact Weyl-parallel manifolds (with A. Derdzinski)
    Monatshefte für Mathematik, vol. 203 (2024), no. 4, pp. 859-871.
    [Journal]-[arXiv]-[MathSciNet]
    
    Abstract

    We prove the existence of compact pseudo-Riemannian manifolds with parallel Weyl tensor which are neither conformally flat nor locally symmetric, and represent all indefinite metric signatures in all dimensions $n\geq 5$. Until now such manifolds were only known to exist in dimensions $n = 3j + 2$, where $j$ is any positive integer [10]. As in [10], our examples are diffeomorphic to nontrivial torus bundles over the circle and arise from a quotient-manifold construction applied to suitably chosen discrete isometry groups of diffeomorphically-Euclidean "model" manifolds

Books

1. Introduction to Lorentz Geometry: Curves and Surfaces (with A. Lymberopoulos), Chapman and Hall/CRC Press, Boca Raton, FL, 2021. ix+340 pp. Book cover displayed on the left.
2. Introdução à Geometria Lorentziana: Curvas e Superfícies (with A. Lymberopoulos), Brazilian Mathematical Society, Universitary Texts Collection, vol. 21, Rio de Janeiro, RJ, 2018. 546 pp.
   In Portuguese. We have a support page for the book.

Scientific dissemination and other texts

1. Corrections of minor misstatements in several papers on ECS manifolds (with A. Derdzinski), e-print arXiv:2404.09766 (not intended for publication).
2. Mergulhos Clássicos de Variedades Grassmannianas: uma visão geral, Revista Matemática Universitária, vol. 1 (2021), pp. 1-14. (In Portuguese.)
3. Topics in Lorentz Geometry, e-print arXiv:1908.01710, 2019.
4. Usando Geometria Diferencial para classificar trajetórias de fótons na Relatividade Especial, Acta Legalicus (ICMC-USP), no. 14 (2018), 14 pp. (In Portuguese.)

Some slides/notes for talks, etc.

• A magnetic version of E. Hopf's theorem (AMS Special Session on Metric Geometry and Topology II, JMM 2025).
• Codazzi tensors in homogeneous spaces (GSTGC 2024).
• Killing vector fields on compact pseudo-Kähler $\partial\bar{\partial}$ manifolds are holomorphic. (Special Holonomy and Geometric Structures on Complex Manifolds, 2024).
• Compact locally homogeneous manifolds with parallel Weyl curvature (Symmetry and Geometry in South Florida, 2024).
• The topology of compact Lorentzian manifolds with parallel Weyl curvature (XI International Meeting on Lorentzian Geometry).
• On compact Cotton-parallel three-manifolds (AMS Special Session on Metric Geometry and Topology II, JMM 2024).
• An overview of completeness in Lorentzian geometry (Oklahoma State University MGSS seminar, 2023).
• Compactifying rank-one Weyl-parallel manifolds (GTA Philadelphia 2023).
• On compact rank-one ECS manifolds (2023 Midwest Geometry Conference).
• Magnetic cotangent bundles (2022 Midwest Dynamical Systems Early Career Conference).
• On ridigity of $0$-isotropic submanifolds of Lorentzian space forms (X International Meeting on Lorentzian Geometry).
• Contrasts between Riemannian and Lorentzian Geometry (First year anniversary of the undergraduate Mathematics program at Federal Institute of Ceará, 2020). In Portuguese.