Symmetric spaces seminar

Due to their high degree of symmetry, symmetric spaces allow one to transfer many problems related to their geometry to the simpler level of Lie groups and Lie algebras. As a consequence, they are a perfect test ground for a wide variety of questions in differential geometry.

This online seminar focuses primarily on isometric actions on symmetric spaces and submanifold theory thereof. We will sometimes take some diversions into adjacent areas like various geometric structures on symmetric spaces, homogeneous spaces, cohomogeneity one manifolds, polar actions, certain topics in Lie theory, etc. The seminar is oriented towards young researchers in the field, especially PhDs and postdocs, but everyone is welcome to join!

Talks usually take place on Wednesdays at 12:00-13:00 GMT, via Zoom.

If you would like to receive talk announcements as well as Zoom access links (usually sent out on Mondays), please email me (ivan.solonenko [at] kcl.ac.uk).

Most talks are recorded and uploaded to the YouTube channel of the seminar.

Past talks

Spring term 2021-22

  • 4 May 2022: José Carlos Díaz-Ramos (University of Santiago de Compostela)
    Topic: Cohomogeneity one actions on quaternionic hyperbolic spaces
    Abstract: In this talk I will present the classification procedure to obtain the classification of cohomogeneity one actions on quaternionic hyperbolic spaces, as well as some byproducts of this study, such as the existence of inhomogeneous isoparametric hypersurfaces with constant principal curvatures.
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  • 20 April 2022: Miguel Domínguez-Vázquez (University of Santiago de Compostela)
    Topic: Polar foliations on symmetric spaces
    Abstract: A polar foliation is a decomposition of a Riemannian manifold into equidistant submanifolds (called leaves) of possibly different dimensions, and such that through any point there exists a submanifold (called section) intersecting all leaves perpendicularly. These objects arise as generalizations of important concepts, such as isoparametric hypersurfaces and polar actions, whose study has produced many beautiful and profound results over the last decades. In this expository talk I will present an introduction to polar foliations and isoparametric hypersurfaces in symmetric spaces, and report on some results concerning their classification problem.
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  • 6 April 2022: Yuri Nikolayevsky (La Trobe University, Melbourne)
    Topic: Einstein hypersurfaces in irreducible symmetric spaces
    Abstract: In this talk, I will present the results of the joint paper of Jeong Hyeong Park and myself in which we give a classification of Einstein hypersurfaces in irreducible symmetric spaces. The main theorem states that there are three classes of such hypersurfaces, belonging to three very different "geometries": homogeneous geometry, Legendrian geometry and affine geometry. I will give a brief introduction into these three geometries and explain how they fit together in our classification.
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  • 9 March 2022: Yuji Kondo (Hiroshima University)
    Topic:     A classification of left-invariant pseudo-Riemannian metrics on some nilpotent Lie groups
    Abstract: It is known that a connected and simply-connected Lie group admits only one left-invariant Riemannian metric up to scaling and isometry if and only if it is isomorphic to the Euclidean space, the Lie group of the real hyperbolic space, or the direct product of the three dimensional Heisenberg group and the Euclidean space of dimension n-3.

In this talk, I am going to talk about a classification of left-invariant pseudo-Riemannian metrics of an arbitrary signature for the third Lie groups with n>3 up to scaling and automorphisms. This completes the classifications of left-invariant pseudo-Riemannian metrics for the above three Lie groups up to scaling and automorphisms.

Our classification can be obtained by a certain isometric action of cohomogeneity zero on a pseudo-Riemannian symmetric space. This action is not proper, in fact there exists a non-closed orbit, which allows us to consider degenerations of orbits. Finally, I explain that degenerations of orbits can have an implication with respect to curvature properties.

This study is based on the joint work with Hiroshi Tamaru from Osaka City University (arXiv:2112.09430 [math.DG]).
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  • 2 March 2022: Alberto Rodríguez-Vázquez (University of Santiago de Compostela)
    Topic:     Totally geodesic submanifolds and Hermitian symmetric spaces
    Abstract: Totally geodesic submanifolds in symmetric spaces are those submanifolds with the simplest geometry and they admit a nice algebraic characterization in terms of Lie triple systems. A special class of symmetric spaces where totally geodesic submanifolds can be studied is that of Hermitian symmetric spaces. In these spaces we can use the notion of Kähler angle to measure how a submanifold fails to be complex.

In this talk, I will report on an ongoing work where a method to construct totally geodesic submanifolds with non-trivial constant Kähler angle in non-flat Hermitian symmetric spaces is given.
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  • 16 February 2022: Tomás Otero Casal (University of Santiago de Compostela)
    Topic:     Symmetric spaces of noncompact type III: Parabolic subgroups and subalgebras
    Abstract: Parabolic subgroups are of special interest when studying isometric actions on symmetric spaces of noncompact type. Geometrically speaking, given a symmetric space of noncompact type M=G/K, proper parabolic subgroups of G are the stabilizers of points at infinity of M. Throughout this third talk, we will introduce parabolic subalgebras from both the geometric and algebraic viewpoints and explain how one can construct "standard" parabolic subalgebras from a choice of a subset of simple roots.

We will also talk about some decomposition results for parabolic Lie subalgebras and the decompositions they induce at the group/manifold level. In particular, this will allow us to introduce a special type of totally geodesic submanifolds of M called boundary components, which are symmetric spaces behaving in an especially nice way with respect to the root space decomposition.
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  • 9 February 2022: Ivan Solonenko (King's College London - LSGNT)
    Topic:     Symmetric spaces of noncompact type II: Roots and Dynkin diagrams
    Abstract: Last week we discussed some basic geometric properties of symmetric spaces of noncompact type in the context of the Cartan and Iwasawa decompositions. This time, we will look more closely at the algebraic side of the picture. We will dive in greater detail into the restricted root space decomposition of the isometry Lie algebra of a symmetric space of noncompact type and study its main properties. There are two stark distinctions between this decomposition and the – probably more well-known – root space decomposition of a complex semisimple Lie algebra: restricted root systems may be nonreduced, while restricted root subspaces may have dimension greater than 1. Somewhat surprisingly, by allowing a root system to be nonreduced, one only gets one additional series in the classification of irreducible root systems, namely the nonreduced system (BC)_r. We will examine the classification of noncompact symmetric spaces through the lens of restricted root systems and their Dynkin diagrams. Since irreducible noncompact symmetric spaces are essentially in one-to-one correspondence with simple noncompact real Lie algebras, this will give us a nice perspective on the classification of the latter. Finally, we will investigate the Weyl and automorphism groups of the restricted root system and observe how they can be thought of in terms of isometries of the underlying space.
    Notes: Slides are available.
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  • 2 February 2022: Juan Manuel Lorenzo Naveiro (University of Santiago de Compostela)
    Topic: Symmetric spaces of noncompact type I: The Cartan and Iwasawa decompositions
    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.

References:

- J. C. Díaz-Ramos, M. Domínguez-Vázquez, and V. Sanmartín-López, “Submanifold geometry in symmetric spaces of noncompact type”, São Paulo Journal of Mathematical Sciences, vol. 15, no. 1, pp. 75–110, 2021.
- S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Academic Press, 1978.
- A. Knapp, Lie Groups Beyond an Introduction (Progress in Mathematics), 2nd edn. Birkhäuser, 2002, vol. 140.
- B. O’Neill, Semi-Riemannian geometry with applications to relativity. Academic press, 1983.
Notes: Slides are available.
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  • 26 January 2022: Ivan Solonenko (King's College London - LSGNT)
    Topic: The index of symmetry of a Riemannian manifold
    Abstract: Riemannian symmetric spaces constitute arguably the most well-understood class of Riemannian manifolds, mainly because they can be extensively studied and ultimately classified by means of Lie theory. To an arbitrary Riemannian manifold, one can naturally assign a number, called the index of symmetry, which measures how badly the manifold fails to be a symmetric space. If the manifold is complete, it admits a natural (in general, singular) foliation, called the foliation of symmetry, which is invariant under isometries and such that the index of symmetry of the manifold is precisely the dimension of the smallest leaf. What is more, each leaf turns out to be a symmetric space in the induced metric. In case the manifold is homogeneous, its foliation of symmetry has all its leaves of the same dimension and thus becomes a genuine foliation. Although its construction is very natural and intrinsic, the foliation (and thus the index) of symmetry is fairly hard to compute even for homogeneous spaces.

In this talk, I will define the foliation of symmetry and introduce its main properties and then tell about some situations when it can be explicitly computed, namely for normal compact homogeneous spaces and a certain class of compact naturally reductive spaces. I will also present the classification of compact homogeneous spaces with a sufficiently high index of symmetry. I will follow these papers by Olmos, Reggiani, Tamaru, and Berndt: [1], [2].
Notes: Slides are available.
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Fall term 2021-22

  • 15 December 2021: Luis Pedro Castellanos Moscoso (Osaka City University)
    Topic: Moduli spaces and left-invariant symplectic structures on Lie groups
    Abstract: In the setting of Lie groups, it is natural to ask about the existence of left-invariant structures. A symplectic Lie group is a Lie group endowed with a left-invariant symplectic form. There are interesting results on the structure of symplectic Lie groups and some classifications in low dimensions, but the general picture is far from complete.

In this talk we present an approach to classify left-invariant symplectic structures on Lie groups. The procedure is based on the moduli space of left-invariant nondegenerate 2-forms, which is a certain orbit space in the set of all nondegenerate 2-forms on a Lie algebra. We present some of the results obtained so far with this approach, including a classification of left-invariant symplectic structures on some almost abelian Lie algebras.
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  • 1 December 2021: Juan Manuel Lorenzo Naveiro (University of Santiago de Compostela)
    Topic: Introduction to Polar Actions
    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.
Notes: Slides are available.