Homogeneous descriptions and families of homogeneous structures.

• Autores: José Luis Carmona Jiménez
• Directores de la Tesis: Marco Castrillón López (dir. tes.)
• Lectura: En la Universidad Complutense de Madrid ( España ) en 2024
• Idioma: inglés
• Títulos paralelos:
  • Descripciones homogéneas y familias de estructuras homogéneas
• Tribunal Calificador de la Tesis: Luis Giraldo Suárez (presid.) , María Pe Pereira (secret.) , Anna Fino (voc.) , Luis Guijaro Santamaria (voc.) , M. Eugenia Rosado María (voc.)
• Enlaces
  • Tesis en acceso abierto en: Docta Complutense
• Resumen

  • Homogeneous spaces are, due to their simplicity, the favorite objects of study for geometers and physicists. Local properties often extend to global ones. Undoubtedly, these spaces are among the most extensively researched and give rise to the first examples of many new theories. These include Euclidean spaces, spheres, hyperbolic spaces, and Lie groups, all classified as homogeneous spaces.

    Homogeneous spaces are differentiable manifolds where there is a transitive action of a Lie group. That is, a group of global transformations such that for any two points, there exists a transformation that sends one point to the other. Under suitable conditions and applying those transformations, we can transport any tensor we have at one point to another point, for example, a metric or a symplectic tensor. In particular, if a metric is present, the homogeneous space is called Riemannian.

    The Ambrose-Singer Theorem characterizes Riemannian homogeneous spaces via the existence of an invariant tensor that satisfies a system of covariant equations. This tensor is called a homogeneous structure and the Ambrose-Singer Theorem is the cornerstone of the program initiated by Tricerri-Vanhecke that studies Riemannian homogeneous manifolds through their homogeneous structures.

    This program establishes the following principles: To characterize a specific Riemannian homogeneous space is to know all its homogeneous structures; Homogeneous structures can be used to differentiate transitive actions.

    In this thesis, the object of study is homogeneous spaces, and we apply techniques derived from the Tricerri-Vanhecke program. Describing all homogeneous structures of a specific homogeneous space is, in general, a challenging problem. In fact, many homogeneous structures of most Riemannian homogeneous spaces remain unknown. 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.

    Many generalizations of the Ambrose-Singer Theorem and the Tricerri-Vanhecke program have been discovered over the years. However, all of them share two common characteristics: There must exist a metric tensor and they only apply to transitive actions. Following that philosophy, the second part of this thesis aims at tackling and weakening those two conditions.

    In reductive homogeneous spaces, we generalize the Ambrose-Singer Theorem with the presence of a finite set of invariant tensors, eliminating the requirement of a metric. We provide a broader definition of homogeneous structure, and new tensors assume significance in the overarching theory. Finally, we apply those results to symplectic homogeneous spaces.

    In non-transitive actions, we focus on Riemannian cohomogeneity one manifolds. That is, there is a group of transformations acting on the manifold in such a way that the orbits of the action are hypersurfaces. This is the closest case to a transitive action, without being transitive. In this framework, 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. Then, we apply this to study cohomogeneity one actions on Euclidean spaces and real hyperbolic spaces.