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José

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I am a mathematician with a Ph.D. in Differential Geometry from the Complutense University of Madrid (2024), where I worked under the supervision of Professor Marco Castrillón López. My doctoral thesis focused on homogeneous and locally homogeneous spaces, Lie group actions, and invariant connections. My research has advanced the understanding of holonomy and the Ambrose–Singer Theorem, with publications in good journals such as Transformation Groups, Annals of Global Analysis and Geometry, and Mediterranean Journal of Mathematics.

Currently, I hold a postdoctoral position at the Simion Stoilow Institute of Mathematics of the Romanian Academy (IMAR), funded by the European Union’s PNRR program, where I collaborate with Professors Andrei and Sergiu Moroianu on the project Conformal Aspects of Geometry and Dynamics. My recent contributions (in 2025) include new results on cohomogeneity one manifolds [CCD25], canonical reductive decompositions of extrinsic homogeneous submanifolds [CC25], and Weyl structures reducible in the direction of the Lee form [Car25].

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Research interest

Due to their elegance and the insights they offer into the interaction between symmetry and geometry, I am interested in group and pseudo-group actions. In particular, I am specialized in transitive actions, that is, the manifold is homogeneous or locally homogeneous. My general goal is to describe the insights of group actions preserving tensor structures through the study of solutions of a system of covariant derivatives. Specially, I aim to study Ambrose-Singer connections.

In parallel, I am also interested in conformal geometry, with a special focus on Weyl connections, as they provide a natural framework to study curvature, holonomy, and invariants beyond the Riemannian setting.

  • Name: José Luis
  • Age: 29
  • From: Spain
  • Email: jcarmona@imar.ro
  • Research Institute: Simion Stoilow Institute of Mathematics of the Romanian Academy
  • Address: 21 Calea Grivitei Street, 010702 Bucharest, Romania

Articles and manuscripts

[Car25] Preprint (ArXiv)

On Weyl structures reducible in the direction of the Lee form

J.L. Carmona Jiménez

A Weyl structure on a Riemannian manifold $(M,g)$ is a torsion-free linear connection $\nabla$ such that there is a $1$-form $\theta$ (called the Lee form) satisfying $\nabla g = 2\, \theta \otimes g$. We examine the case in which there exists a $\nabla$-parallel distribution of codimension $1$ on which the Lee form vanishes identically. We prove that if $(M,g)$ is complete with $\theta$ closed, then the Weyl structure must be flat or exact. We apply this to show that every homogeneous Kenmotsu manifold is isometric to the real hyperbolic space.

arXiv
[CCD25] Transformation Groups (2025)

The Ambrose-Singer Theorem for Cohomogeneity one Riemannian manifolds

J.L. Carmona Jiménez, M. Castrillón López, J. C. Díaz Ramos

We characterize isometric actions whose principal orbits are hypersurfaces through the existence of a linear connection satisfying a set of covariant equations in the same spirit as the Ambrose-Singer Theorem for homogeneous space. These results are then used to describe isometric cohomogeneity one foliations in terms of such connections. Finally, we provide explicit examples of these objects in Euclidean spaces and real hyperbolic spaces.

DOI: 10.1007/s00031-025-09927-x arXiv
[CC25] Preprint (ArXiv)

Canonical Reductive Decomposition of Extrinsic Homogeneous Submanifolds

J.L. Carmona Jiménez, M. Castrillón López

Let $\overline{M}=\overline{G}/\overline{H}$ be a homogeneous Riemannian manifold. Given a Lie subgroup $G\subset \overline{G}$ and a reductive decomposition of the homogeneous structure of $\overline{M}$, we analyze a canonical reductive decomposition for the orbits of the action of $G$. These leaves of the $G$-action are extrinsic homogeneous submanifolds and the analysis of the reductive decomposition of them is related with their extrinsic properties. We connect the study with works in the literature and initiate the relationship with the Ambrose-Singer theorem and homogeneous structures of submanifolds.

arXiv
PhD Thesis (2024)

Homogeneous descriptions and families of homogeneous structures

J.L. Carmona Jiménez

In this thesis, the object of study is homogeneous spaces, and we apply techniques derived from the Tricerri-Vanhecke program. First, we characterize all the homogeneous structures of the complex hyperbolic space. Afterwards, we examine the process of reduction of homogeneous structures when the fibre of the reduction is one-dimensional. We turn to a non-metric framework and we generalize the Ambrose-Singer Theorem with the presence of a finite set of invariant tensors, eliminating the requirement of a metric. Moreover, we apply those results to symplectic homogeneous spaces. We finish with non-transitive actions, we prove that the existence of a Riemannian cohomogeneity one action is equivalent to the existence of a tensor satisfying a system of covariant equations.

PDF Slides of the Defense
[CC22] Mediterr. J. Math. 19, 280 (2022).

The Ambrose–Singer Theorem for General Homogeneous Manifolds with Applications to Symplectic Geometry.

J.L. Carmona Jiménez, M. Castrillón López

The main Theorem of this article provides a characterization of reductive homogeneous spaces equipped with some geometric structure (not necessarily pseudo-Riemannian) in terms of the existence of certain connection. This result generalizes the well-known Theorem of Ambrose and Singer for Riemannian homogeneous spaces (Ambrose and Singer in Duke Math J 25(4):647–669, 1958). We relax the conditions in this theorem and prove a characterization of reductive locally homogeneous manifolds. Finally, we apply these results to classify, with explicit expressions, reductive locally homogeneous almost symplectic, symplectic and Fedosov manifolds.

DOI: 10.1007/s00009-022-02197-x arXiv
[CC22*] Ann. Glob. Anal. Geom. 62, (2022)

The homogeneous holonomies of complex hyperbolic space.

J.L. Carmona Jiménez, M. Castrillón López

We describe the holonomy algebras of all canonical connections and their action on complex hyperbolic spaces $\mathbb{C}\mathrm{H}(n)$ in all dimensions ($n\in\mathbb{N}$). This thorough investigation yields a formula for all Kähler homogeneous structures on complex hyperbolic spaces. Finally, we have related the belonging of the homogeneous structures to the different Tricerri and Vanhecke’s (or Abbena and Garbiero’s) orthogonal and irreducible $\mathrm{U}(n)$-submodules with concrete and determined expressions of the holonomy.

DOI: 10.1007/s10455-022-09852-2 arXiv
[CC20] Axioms, 9(3):94 (2020).

Reduction of Homogeneous Pseudo-Kähler Structures by One-Dimensional Fibers

J.L. Carmona Jiménez, M. Castrillón López

We study the reduction procedure applied to pseudo-Kähler manifolds by a one dimensional Lie group acting by isometries and preserving the complex tensor. We endow the quotient manifold with an almost contact metric structure. We use this fact to connect pseudo-Kähler homogeneous structures with almost contact metric homogeneous structures. This relation will have consequences in the class of the almost contact manifold. Indeed, if we choose a pseudo-Kähler homogeneous structure of linear type, then the reduced, almost contact homogeneous structure is of linear type and the reduced manifold is of type $\mathcal{C}_5 \oplus \mathcal{C_6} \oplus \mathcal{C}_{12}$ of Chinea-González classification.

DOI: 10.3390/axioms9030094