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Hello I'm

José

Welcome to my personal place. I hold a Ph.D. in Mathematics with a specialization in general homogeneous spaces, which involves a Lie group acting transitively on a manifold, under the supervision of M. Castrillón López at University Complutense of Madrid. In my published papers, I have explored transitive actions in various contexts, utilizing a range of techniques.

Additionally, I have had the opportunity to undertake two research stays, one in Santiago de Compostela (Spain) and another in Marburg (Germany). During these stays, I collaborated with professors such as J.C. Díaz-Ramos and I. Agricola, whose contributions to our research area are invaluable. I was privileged to work alongside their outstanding research groups.

Currently, I am engaged in a postdoctoral research position at the Simion Stoilow Institute of Mathematics of the Romanian Academy in Bucharest. Here, I continue my work in homogeneous spaces and collaborate closely with professors S. Moroianu and A. Moroianu. I am eager to explore new avenues of research and foster potential collaborations in the future.

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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.

  • Name: José Luis
  • Age: 28
  • 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

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
Preprint (ArXiv)

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.

arXiv
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
Ann. Glob. Anal. Geom. 62, 391–411 (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
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