Research articles

• J. M. Lorenzo-Naveiro, A. Rodríguez-Vázquez
  Totally geodesic submanifolds of the homogeneous nearly Kähler 6-manifolds
  and their \(\mathsf{G}_{2}\)-cones
  arxiv:2411.11261
  Abstract.

  In this article we classify totally geodesic submanifolds of homogeneous nearly Kähler 6-manifolds, and of the \(\mathsf{G}_{2}\)-cones over these 6-manifolds. To this end, we develop new techniques for the study of totally geodesic submanifolds of analytic Riemannian manifolds, naturally reductive homogeneous spaces and Riemannian cones. In particular, we obtain an example of a totally geodesic submanifold with self-intersections in a simply connected homogeneous space.
• J. C. Díaz-Ramos, J. M. Lorenzo-Naveiro
  Codimension two polar homogeneous foliations on symmetric spaces of noncompact type
  arxiv:2302.08339
  Abstract.

  We classify homogeneous polar foliations of codimension two on irreducible symmetric spaces of noncompact type up to orbit equivalence. Any such foliation is either hyperpolar or the canonical extension of a polar homogeneous foliation on a rank one symmetric space.
• I. Solonenko, J. M. Lorenzo-Naveiro
  Sections of polar actions
  arXiv:2111.05280
  Abstract.

  In this short note we provide an elementary proof of the folklore result in the theory of isometric Lie group actions on Riemannian manifolds asserting that sections of polar actions are totally geodesic.

Work in progress

• J. M. Figueroa-O'Farrill, J. M. Lorenzo-Naveiro
  Coisotropy-one homogeneous spacetimes I: (3,2) kinematical Lie algebras

Book chapters

• La geometría del grupo de Heisenberg (in Spanish)
  As matemáticas do veciño. Actas do Seminario de Iniciación á Investigación (SII)
  Instituto de Matemáticas, Universidade de Santiago de Compostela (2021). ISSN: 2171-6536

Monographs

• Foliaciones polares homogéneas en espacios simétricos (in Spanish)
  Publications of the Department of Geometry and Topology 149, Universidade de Santiago de Compostela (2021).
  This is my Master's Thesis, in which I classify polar homogeneous foliations in \(\mathsf{SL}(3,\mathbb{R})/\mathsf{SO}(3)\)
• Grupos de transformaciones (in Spanish)
  Minerva, Repositorio Institucional da USC (2021).
  This is my undergraduate thesis, in which I describe the basic theory of transformation groups in differential geometry

Talks at conferences

• Totally geodesic submanifolds of nearly Kähler and \(\mathsf{G}_{2}\)-manifolds, Meeting of young researchers of the Spanish Network of Geometric Analysis (2024), Universidade de Santiago de Compostela, 28th November 2024.
  Abstract (in Spanish).

  Una variedad casi hermitiana \((M^{2n},J)\) se dice nearly Kähler si la derivada covariante \(\nabla J\) es totalmente antisimétrica. Dada una 6-variedad (estrictamente) nearly Kähler simplemente conexa \(M\neq\mathsf{S}^{6}\), es posible reescalar la métrica de \(M\) de tal modo que su cono \(\widehat{M}\) tiene holonomía especial \(\mathsf{G}_{2}\). Un teorema de Butruille afirma que las variedades simplemente conexas, homogéneas, y estrictamente nearly Kähler de dimensión 6 son \(\mathsf{S}^{6}\), el espacio proyectivo \(\mathbb{C}\mathsf{P}^{3}\), la variedad de banderas \(\mathsf{F}(\mathbb{C}^{3})\) y el casi producto \(\mathsf{S}^{3}\times\mathsf{S}^{3}\). Estas caen dentro de la clase de espacios homogéneos naturalmente reductivos, cuya geometría se puede entender puramente en términos de álgebras de Lie.
  El objetivo de esta charla es describir la clasificación de las subvariedades totalmente geodésicas de los espacios anteriormente mencionados, así como de sus conos riemannianos. Para ello, introduciremos las herramientas necesarias para poder trabajar con espacios homogéneos naturalmente reductivos y conos, para mías adelante ilustrar los ejemplos que aparecen en cada caso.
  Esta charla está basada en un trabajo conjunto con Alberto Rodríguez Vázquez (Université Libre de Bruxelles)
• Nearly Kähler geometry and totally geodesic submanifolds, Symmetry and Shape (2024), Universidade de Santiago de Compostela, 24th September 2024.
  Abstract.

  An almost Hermitian manifold \((M^{2n},J)\) is said to be nearly Kähler if the covariant derivative of \(J\) is totally skew-symmetric. It can be shown that if \(M\neq \mathsf{S}^{6}\) is a simply connected six-dimensional strict nearly Kähler manifold, then one can rescale the metric on \(M\) so that its Riemannian cone \(\widehat{M}\) has special holonomy \(\mathsf{G}_{2}\). A theorem of Butruille asserts that the simply connected homogeneous strict nearly Kähler manifolds of dimension six are the round sphere \(\mathsf{S}^{6}\), the space \(\mathsf{F}(\mathbb{C}^{3})\) of full flags in \(\mathbb{C}^{3}\), the complex projective space \(\mathbb{C}\mathsf{P}^{3}\) and the almost product \(\mathsf{S}^{3}\times\mathsf{S}^{3}\). These spaces belong to the general class of naturally reductive homogeneous spaces, whose geometry can be understood in purely Lie-algebraic terms.
  The purpose of this talk is to discuss the classification of totally geodesic submanifolds of the (non-symmetric) homogeneous strict nearly Kähler 6-manifolds, as well as their \(\mathsf{G}_{2}\) cones. To this end, we will develop the algebraic framework needed to attack the problem, and later on we will exhibit the examples that appear in each case. This is a joint work with Alberto Rodríguez-Vázquez (KU Leuven).
• Polar homogeneous foliations on symmetric spaces of noncompact type, Differential Geometry and its Applications (2022), University of Hradec Králové (Czech Republic), 18th July 2022.
  Recording of the Talk
  Abstract.

  An isometric action of a Lie group on a Riemannian manifold is polar if there exists a (totally geodesic) submanifold meeting all orbits orthogonally. These actions generalize common situations arising in different areas of Mathematics, such as the spectral theorem for self adjoint matrices or the polar coordinate system on \(\mathbb{R}^{2}\). The aim of this talk is to discuss some results on polar actions without singular orbits on symmetric spaces of noncompact type whose isometry Lie algebra \(\mathfrak{g}\) is split. The deep connection between noncompact symmetric spaces and real semisimple Lie algebras allows us to study such actions from an algebraic point of view [1]. By a theorem of Mostow on maximal solvable subalgebras of semisimple Lie algebras [2], the problem is reduced to finding subalgebras of the solvable part of g with respect to the Iwasawa decomposition satisfying certain properties involving their Lie bracket. Finally, we will present the classification of polar homogeneous foliations on \(\mathsf{SL}(3,\mathbb{R})/\mathsf{SO}(3)\), the space of volume preserving inner products on \(\mathbb{R}^{3}\).
  References:
  [1] J. Berndt, J. C. Díaz-Ramos, H. Tamaru: Hyperpolar homogeneous foliations on symmetric spaces of noncompact type. J. Diff. Geom. 86 (2010), 191–235.
  [2] G. D. Mostow: On maximal subgroups of real Lie groups. Ann. of Math. (2) 74 (1961), 503–517.
  Note: Slides are avaliable.
• Polar actions on some noncompact symmetric spaces, Workshop on Manifolds with Symmetries (2022), University of Stuttgart (Germany), 29th March 2022.
  
  Abstract.

  An isometric action on a Riemannian manifold is said to be polar if it admits sections, that is, submanifolds that meet every orbit perpendicularly. Such actions provide a generalization to common situations arising in geometry, such as the system of polar coordinates on the plane or the spectral theorem for symmetric matrices. As of today, it is an open problem to classify polar actions on several families of ambient manifolds.
  During this talk, we will focus on the classification problem on symmetric spaces of noncompact type. The deep connection between the theory of symmetric spaces and real semisimple Lie groups allows us to reformulate the geometric properties of polar actions in almost purely algebraic terms. With this relationship in mind, we will discuss the problem of determining all polar actions on \(\mathsf{SL}(3,\mathbb{R})/\mathsf{SO}(3)\), the space of volume preserving positive definite linear maps over \(\mathbb{R}^{3}\).
  Note: Slides are available.
• Polar homogeneous foliations on \(\mathsf{SL}(3,\mathbb{R})/\mathsf{SO}(3)\), Symmetry and Shape (2021), Universidade de Santiago de Compostela (Spain), 18th October 2021.
  
  Abstract.

  An isometric action of a Lie group over a Riemannian manifold is said to be polar if there exists a submanifold that meets all orbits perpendicularly. Such a submanifold is known as a section, and is totally geodesic. As of today, it is an open problem to classify these actions (up to orbit equivalence) in several families of Riemannian manifolds, mainly symmetric spaces.
  The aim of this talk will be to focus on polar actions without singular orbits when the ambient manifold is a symmetric space of noncompact type. In this setting, the existence of a section can be characterized by means of an algebraic criterion [1]. Combining this with the work of Mostow on maximal solvable subalgebras of real semisimple Lie algebras [2], we will arrive at the complete classification of all polar homogeneous foliations on \(\mathsf{SL}(3,\mathbb{R})/\mathsf{SO}(3)\), the space of volume-preserving, self-adjoint and positive definite linear transformations of the Euclidean \(3\)-space.
  Bibliography:
  [1] J. Berndt, J. C. Díaz-Ramos, H. Tamaru: Hyperpolar homogeneous foliations on symmetric spaces of noncompact type, J. Diff. Geom. 86 (2010), 191-235.
  [2] G. D. Mostow: On maximal subgroups of real Lie groups, Ann. of Math. (2) 74 (1961) 503-517.

Talks at seminars

• Totally geodesic submanifolds of nearly Kähler and \(\mathsf{G}_{2}\)-manifolds, Seminar of the Institute of Geometry and Topology, Universität Stuttgart (Germany), 14 November 2024.
  
  Abstract.

  An almost Hermitian manifold is said to be nearly Kähler if the covariant derivative of its almost complex structure is totally skew-symmetric.
  Im dimension six, these spaces are of particular interest, as the Riemannian cone of a six-dimensional strict nearly Kähler manifold (except for the sphere \(\mathsf{S}^{6}\)) has special holonomy \(\mathsf{G}_{2}\). It was shown by Butruille that the simply connected homogeneous strict nearly Kähler manifolds of dimension six are \(\mathsf{S}^{6}\), the flag manifold \(\mathsf{F}(\mathbb{C}^{3})\), the complex projective space \(\mathbb{C}\mathsf{P}^{3}\) and the almost product \(\mathsf{S}^{3}\times\mathsf{S}^{3}\).
  In this talk, I will report on a joint work with Alberto Rodríguez Vázquez in which we classify the totally geodesic submanifolds of the (non-symmetric) homogeneous nearly Kähler 6-manifolds, as well as their \(\mathsf{G}_{2}\)-cones. To this end, we develop the tools to attack the problem in (naturally reductive) homogeneous spaces and Riemannian cones.
• Polar actions on symmetric spaces of noncompact type, Pure mathematics seminar, University of Queensland (Australia), 19 July 2024.
  
  Abstract.

  An isometric action of a Lie group on a complete Riemannian manifold is said to be polar if there exists a submanifold that intersects every orbit orthogonally. Such a submanifold is known as a section and is totally geodesic. These actions give a generalization of several well-known concepts in geometry, such as the polar coordinate system in the Euclidean plane or the spectral theorem for self-adjoint operators.
  The aim of this talk is to give an overview of the classification problem for polar actions on symmetric spaces, focusing on those of noncompact type. Later, I will report on a joint work with J.C. Díaz-Ramos (Universidade de Santiago de Compostela) in which we classify polar homogeneous foliations on symmetric spaces of noncompact type whose section is homothetic to the hyperbolic plane. To this end, we will describe a method proposed by J. Berndt and H. Tamaru which allows us to extend submanifolds and isometric actions on hyperbolic spaces to arbitrary symmetric spaces, and show how the aforementioned foliations can be obtained from this procedure.
• Nearly Kähler geometry and totally geodesic submanifolds, Differential Geometry Seminar, University of Adelaide (Australia), 31st May 2024.
  
  Abstract.

  A theorem of Butruille asserts that the (simply connected, homogeneous) Riemannian manifolds of dimension six admitting a strict nearly Kähler metric are the round sphere \(\mathsf{S}^{6}\), the space \(\mathsf{F}(\mathbb{C}^{3})\) of full flags in \(\mathbb{C}^{3}\), the complex projective space \(\mathbb{C}\mathsf{P}^{3}\) and the almost product \(\mathsf{S}^{3}\times\mathsf{S}^{3}\). These spaces belong to the general class of naturally reductive homogeneous spaces, whose geometry can be understood in purely Lie-algebraic terms.
  The aim of this talk is to describe a joint work with Alberto Rodríguez-Vázquez (KU Leuven) in which we classify the totally geodesic submanifolds of the aforementioned spaces. To this end, we will develop the algebraic tools needed to work with naturally reductive homogeneous spaces and to attack our problem, and later on we will exhibit the examples that appear in each case.
• Totally geodesic submanifolds in homogeneous six-dimensional nearly Kähler spaces, Geometry day, Universidade de Santiago de Compostela (Spain), 11th April 2024.
  
  Abstract.

  The (strict) nearly Kähler homogeneous Riemannian manifolds of dimension six have been classified by Butruille. Indeed, these are the round sphere \(\mathsf{S}^{6}\), the flag manifold \(\mathsf{F}(\mathbb{C}^{3})\), the complex projective space \(\mathbb{C}\mathsf{P}^{3}\) and \(\mathsf{S}^{3}\times\mathsf{S}^{3}\). These spaces are endowed with a naturally reductive structure, which allows us to understand their geometric properties by means of Lie algebras. In this talk, I will describe a joint work with Alberto Rodríguez-Vázquez (KU Leuven) in which we classify the totally geodesic submanifolds of the aforementioned spaces. To this end, we will develop the algebraic framework needed to attack the problem, and later on we will exhibit the examples that appear in each case, as well as their relationship with the ambient almost complex structure.
• Totally geodesic submanifolds of the homogeneous nearly Kähler 6-manifolds, Seminar of the Geometry Section, KU Leuven (Belgium), 16th February 2024.
  
  Abstract.

  A theorem of Butruille asserts that the homogeneous strict nearly Kähler Riemannian manifolds of dimension six are the space of full flags in \(\mathbb{C}^{3}\), the complex projective space \(\mathbb{C}\mathsf{P}^{3}\) and \(\mathsf{S}^{3}\times\mathsf{S}^{3}\). These belong to the general class of naturally reductive homogeneous spaces, whose geometry can be understood entirely in terms of Lie algebras.
  The purpose of this talk is to discuss the classification of totally geodesic submanifolds in the three spaces given above. To this end, we will develop the algebraic framework needed to attack the problem, and later on we will exhibit the examples that appear in each case.
  This is a joint work with Alberto Rodríguez-Vázquez (KU Leuven).
• Polar homogeneous foliations on noncompact symmetric spaces, Seminar of the Geometry Section, KU Leuven (Belgium), 19th May 2023.
  
  Abstract.

  An isometric action of a Lie group on a Riemannian manifold is polar if there exists a submanifold intersecting each orbit orthogonally. These actions generalize common situations arising in algebra and geometry, such as the system of polar coordinates in the Euclidean plane and the spectral theorem for symmetric matrices. As of today, it is an open problem to determine all polar actions on Riemannian manifolds with a large group of isometries.
  The aim of this talk is to give an overview on some known classification results concerning polar actions, focusing mainly on symmetric spaces. Later, I will talk about a joint work with J. C. Díaz-Ramos in which we classify all polar homogeneous foliations on symmetric spaces of noncompact type whose section is homothetic to the hyperbolic plane.
• Symmetric spaces of noncompact type I: The Cartan and Iwasawa decompositions, Symmetric spaces seminar (2022) (online seminar), 2nd February 2022
  
  Abstract.

  The main property of symmetric spaces is the fact that they can be described in terms of two Lie groups together with an involution representing the geodesic symmetry at a given point. Because of this, one can extensively study the geometry of these manifolds by means of purely algebraic methods. During the course of this and the next two talks, Tomás Otero, Ivan Solonenko, and I will introduce some of those methods in the context of symmetric spaces of noncompact type.
  Throughout this first talk, we will exhibit two decompositions of the isometry group and its Lie algebra for such manifolds. The first one is the Cartan decomposition (valid for any symmetric space), which gives a way to express some invariants of a symmetric space (connection, curvature, geodesics, etc.) in Lie-theoretic terms. On the other hand, one has the Iwasawa decomposition (only valid in the noncompact setting), which serves as a generalization of the Gram-Schmidt process for the isometries of a noncompact symmetric space and allows to realize the space as a simply connected solvable Lie group with a left-invariant metric.
  Note: Slides are available.
• Introduction to Polar actions, Symmetric spaces seminar (2021) (online seminar), 1st December 2021.
  
  Abstract.

  An isometric action of a Lie group on a complete Riemannian manifold is said to be polar if there exists a submanifold that intersects every orbit orthogonally. Such a submanifold is known as a section and is totally geodesic. These actions give a generalization of several well-known concepts in geometry, such as the polar coordinate system in the Euclidean plane or the Spectral Theorem for self-adjoint operators.
  The aim of this talk is to explore several examples and properties of polar actions, with an application to the theory of real semisimple Lie algebras. From these properties, one can obtain an explicit description of their sections. Afterwards, we will derive an algebraic criterion to determine if a given action is polar when the ambient manifold is a symmetric space of compact (or noncompact) type.
  Note: Slides are available.
• La geometría del grupo de Heisenberg (in Spanish), Introduction to Research Seminar (SII) 2021, Instituto de Matemáticas, Universidade de Santiago de Compostela (Spain), 24th February 2021.
  Abstract (in Spanish).

  El hecho fundamental que distingue la Geometría Subriemanniana de la Geometría de Riemann se encuentra en que solamente estudiamos curvas "admisibles", que siguen unas ciertas direcciones determinadas por una distribución.
  El objetivo de esta charla será dar una introducción a los conceptos principales de la Geometría Subriemanniana, apoyándonos en el grupo de Heisenberg de dimensión \(3\) como ejemplo ilustrativo, cuya construcción está vinculada al problema isoperimétrico en el plano. Trataremos de visualizar las geodésicas y las bolas métricas en este espacio, comentando algunos fenómenos inesperados que surgen de imponer esta condición de admisibilidad.
  Note: Slides are available.

Posters

• Codimension two polar homogeneous foliations on noncompact symmetric spaces, Symmetry and shape (2022), Universidade de Santiago de Compostela (Spain), 14th October 2022 (Poster).
  Abstract.

  An isometric action of a Lie group on a Riemannian manifold is polar if there exists a complete submanifold (called a section) that intersects every orbit perpendicularly. It is an open problem to determine all such actions up to orbit equivalence on ambient manifolds with a large group of isometries (in particular, symmetric spaces).
  We will focus on polar actions without singular orbits on symmetric spaces of noncompact type. In this setting, a complete classification of cohomogeneity one actions is known [2], while in codimension two there is a procedure to construct all possible foliations in which the section is the Euclidean plane, derived from a more general method described in [1]. We will show how to finish the case of codimension two by determining all polar foliations whose section is homothetic to the hyperbolic plane.
  Bibliography:
  [1] J. Berndt, J. C. Díaz-Ramos, H. Tamaru: Hyperpolar homogeneous foliations on symmetric spaces of noncompact type, J. Differential Geom. 86 (2010), 191-235.
  [2] J. Berndt, H. Tamaru, Homogeneous codimension one foliations on noncompact symmetric spaces, J. Differential Geom. 63 (2003), no. 1, 1-40.