Submanifold Geometry and Lie Group Actions 2024

Yoshio Agaoka (Hiroshima University, Professor Emeritus)
Osamu Ikawa (Kyoto Institute of Technology)
Yu Ohno (Hokkaido University)
Taro Kimura (National Institute of Technology, Tsuruoka College)
Isami Koga (Kyushu International University)
Yuichiro Sato (Wasede University, GEC)
Kyoji Sugimoto (Tokyo University of Science)
Kota Hattori (Keio University)
Shinobu Fujii (Chitose Institute of Science and Technology)
Makiko Sumi Tanaka (Tokyo University of Science)

9:45 - 10:45 Yuichiro Sato: "Ricci-flat left-invariant Lorentzian metrics on almost abelian Lie groups"
11:00 - 12:00 Shinobu Fujii: "On Symmetric Clifford systems and subquandles in real Grassmannian manifolds"
13:50 - 14:50 Kyoji Sugimoto: "Antipodal sets of pseudo-Riemannian symmetric $R$-spaces"
15:05 - 16:05 Taro Kimura: "Cartan embeddings of compact Riemannian $k$-symmetric spaces"
16:20 - 17:20 Makiko Sumi Tanaka: "Maximal antipodal subgroups and covering homomorphisms with odd degree"

9:45 - 10:45 Yu Ohno: "Homogeneous structures of 3-dimensional naturally reductive homogeneous spaces"
11:00 - 12:00 Isami Koga: "Equivariant harmonic maps of complex projective spaces"
13:50 - 14:50 Kota Hattori: "The energy of maps accompanying the collapsing of the K3 surface"
15:05 - 16:05 Yoshio Agaoka: "An identity of Ricci curvatures for left invariant metrics --the three dimensional case--"
16:20 - 17:20 Osamu Ikawa: "The intersection of two real flag manifolds in a complex flag manifold, the canonical form of a compact symmetric triad, and their application"

Yoshio Agaoka: "An identity of Ricci curvatures for left invariant metrics --the three dimensional case--"
We explain a partial result on the left invariant version of prescribed Ricci curvature problem. We can easily see that the Ricci curvatures of left invariant metrics on Lie groups satisfy a certain kind of identity, though its explicit form is not known yet. Such an identity gives a necessary condition for a symmetric tensor to be a Ricci curvature of some metric on Lie groups, and so it is an important subject to know its explicit form in the prescribed curvature problem. In this lecture we give the answer of this question for the three dimensional case by giving its explicit form for both solvable and unimodular cases. For the solvable case it can be expressed as a polynomial relation of the Ricci curvature with degree nine, and for the unimodular case it is expressed as a cubic form. To obtain these results, we need some knowledge on invariants of Ricci curvature and structure constants of Lie algebras. We explain in some details on these subjects, including the variety of Lie algebras.

Osamu Ikawa: "The intersection of two real flag manifolds in a complex flag manifold, the canonical form of a compact symmetric triad, and their application"
The necessary and sufficient condition for the intersection of two real flag manifolds in a complex flag manifold to be discrete are stated in a stronger form than previously known. The discrete intersection of real flag manifolds is the orbit of a Weyl group, which is an ‘antipodal set’ of a complex flag manifold. As an application, we show that a real flag manifold in a complex flag manifold is globally tight. The triple of two real flag manifolds and a complex flag manifold as an ambient space is constructed from a compact symmetric triad. It is shown that the compact symmetric triad can be taken to be ‘standard’. This talk is a joint work with O. Shinji, T. Sakai and H. Tasaki. This talk includes the content of joint research with Kurando Baba and that of Hiroshi Iriyeh, Takayuki Okuda, Takashi Sakai and Hiroyuki Tasaki.

Yu Ohno: "Homogeneous structures of 3-dimensional naturally reductive homogeneous spaces"
A naturally reductive homogeneous space is a generalization of Riemannian symmetric spaces and a geodesic through an origin in this space is the orbit of an one-parameter group of isometries. It is known that the dimension of the isotropy group of a 3-dimensional naturally reductive homogeneous space is 6 or 4, and the 3-dimensional naturally reductive homogeneous spaces were completely classified. On the other hand, Ambrose and Singer proved that local homogeneity of a Riemannian manifold is characterized by the existence of the tensor called the homogeneous structure. Moreover in special cases we can classify all the coset representations of a given homogeneous Riemannian manifold by using the homogeneous structures. In this talk we will explain the classification of all the homogeneous structures of 3-dimensional naturally reductive homogeneous spaces whose dimension is 4, and we will determine all the coset representations of these spaces. In particular, there exists a non-trivial coset representation of the universal covering group of the 3-dimensional special linear group $\mathrm{SL}(2,\mathbb{R})$.

Taro Kimura: "Cartan embeddings of compact Riemannian $k$-symmetric spaces"
Let $G$ be a compact connected Lie group, $¥sigma$ be an automorphism of $G$ and put $K=\{g\in G\mid \sigma(g)=g\}$. The homogeneous space $G/K$ is embedded in $G$ by the mapping $\Psi_{\sigma}\,:\, G/K \rightarrow G\,;\, gK\mapsto g\sigma(g^{-1})$, which we call the Cartan embedding of $G/K$ induced from $\sigma$. Also the homogeneous space $G/K$ has the structure of $k$-symmetric space, we note that when $k=2$, $G/K$ is a symmetric space in the usual sense. In this talk, we classify automorphisms of order 3, 4 on compact connected simple Lie groups by which the induced Cartan embedding is an austere, minimal, and biharmonic embedding. This talk is based on joint works with Professor Katsuya Mashimo.

Isami Koga: "Equivariant harmonic maps of complex projective spaces"
In this talk we introduce two results about the construction of moduli spaces of equivariant harmonic maps of complex projective spaces. First, we construct moduli spaces of harmonic maps to complex projective spaces equivariant for symplectic groups. Next, we consider harmonic maps to quaternion projective spaces equivariant for unitary groups. In both cases, we focus on vector bundles equipped with invariant connections in order to apply generalized do Carmo-Wallach Theory. This talk is based on joint work with Yasuyuki Nagatomo (Meiji University).

Yuichiro Sato: "Ricci-flat left-invariant Lorentzian metrics on almost abelian Lie groups"
A Lie group is almost abelian if it has a commutative normal subgroup of codimension one. In this talk, we show a classification theorem for Ricci-flat left-invariant Lorentzian metrics on almost abelian Lie groups. As an application, we introduce the vacuum solution corresponding to a higher dimensional version of the Petrov solution, which is one of the classical solutions in relativity. This talk is based on joint work with Takanao Tsuyuki (Hokkaido Information University).

Kyoji Sugimoto: "Antipodal sets of pseudo-Riemannian symmetric $R$-spaces"
The notion of pseudo-Riemannian symmetric $R$-spaces was introduced by H. Naitoh as a generalization of the notion of symmetric $R$-spaces in 1984. I will show that a maximal antipodal set of the pseudo-Riemannian symmetric $R$-space associated with a semisimple symmetric graded Lie algebra can be obtained as an orbit of a Weyl group.

Kota Hattori: "The energy of maps accompanying the collapsing of the K3 surface"
In this talk, I explain the Dirichlet energy of some smooth maps associated with a collapsing family of hyper-Kaehler metrics on the K3 surface constructed by Foscolo. I introduce an invariant for homotopy classes of smooth maps, showing that it gives a lower bound of the energy. Moreover, when the hyper-Kaehler metrics collapse, the energy converges to the above invariant for Foscolo's collapsing families.

Shinobu Fujii: "On Symmetric Clifford systems and subquandles in real Grassmannian manifolds"
A symmetric Clifford system is a set of finite real symmetric matrices satisfying certain relations, which is known to correspond one-to-one with the representation of the Clifford algebra. It is also known that the symmetric Clifford systems are used to construct isoparametric hypersurfaces of OT--FKM type. Moreover the symmetric Clifford systems correspond to the totally geodesic spheres in certain real Grassmann manifolds, they are also geometrically important objects. In this talk, we define subspace arrangements associated with symmetric Clifford systems, and show that these subspace arrangements generate subquandles of the real Grassmannian manifolds. In addition, we talk about a relation between such arrangements and the maximal s-commutative subsets in the Grassmannian manifolds.

Makiko Sumi Tanaka: "Maximal antipodal subgroups and covering homomorphisms with odd degree"
A compact Lie group G equipped with a bi-invariant Riemannian metric is a Riemannian symmetric space. A subset S of a Riemannian symmetric space is called an antipodal set if the symmetry at each point of S is the identity map on S. A maximal antipodal set of G containing the identity element is a subgroup, called a maximal antipodal subgroup. We show that all of the maximal antipodal subgroups in compact Lie groups which are not necessarily connected do not change through covering homomorphisms with odd degree. This talk is based on joint research with Hiroyuki Tasaki.

Submanifold Geometry and Lie Group Actions 2024

Date: 2-3, December, 2024

Place: Tokyo University of Science, Kagurazaka Campus, Morito Memorial Hall, Forum No. 1

Invited Speakers (confirmed)

Program and Title: PDF(Last Modified: 2024.11.28)

Program and Title

2 December

3 December

Title & Abstract