Submanifold Geometry and Lie Group Actions 2025

Naoki Kato: "Left-invariant transversely affine foliations and a generalization of left-symmetric structures"
It is known that there exists a correspondence between left-invariant affine structures on a simply connected Lie group $G$ and left-symmetric structures on the Lie algebra $\mathfrak{g}$ of $G$. In this talk, we define an algebraic structure on $\mathfrak{g}$, which we call a generalized left-symmetric structure, and give a correspondence between left-invariant transversely affine foliations of $G$ and generalized left-symmetric structures on $\mathfrak{g}$. Moreover, we give an algebraic description of the completeness of left-invariant transversely affine foliations using this correspondence. We also provide examples of low-dimensional Lie algebras which admit left-symmetric structures.

Kazuki Kannaka: "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. When $k=4$, we put $H=\{ g \in G \mid \sigma^2(g)=g \}$. Thus we obtain a fibration $H/K \to G/K \to G/H$. In this talk, We characterize it by means of the symmetric space $G/H$, so that the induced Cartan embedding of compact Riemannian $4$-symmetric spaces is biharmonic. This talk is based on joint works with Professor Katsuya Mashimo.

Taro Kimura: "Hurwitz-Radon number and proper actions of semisimple Lie groups"
The Hurwitz\UTF{2013}Radon number unexpectedly appears in various mathematical fields, including composition laws of quadratic forms and the study of vector fields on spheres. In this talk, I will explain that it also appears in the context of pseudo-Riemannian geometry. Specifically, we will discuss the question: What types of noncompact connected semisimple Lie groups can act isometrically and properly on a given pseudo-Riemannian symmetric space? The talk will begin with an overview of previous research and then explain that, for certain family of pseudo-Riemannian symmetric spaces, such semisimple Lie groups can be classified using the `Hurwitz\UTF{2013}Radon number associated with the classical Lie algebra'. This talk is based on joint work with Koichi Tojo (Tokai university).

Atsumu Sasaki: "Invariant measures on reductive real spherical homogeneous spaces"
For a semisimple symmetric space of a non-compact real semisimple Lie group, the orbit space for the maximal compact subgroup action on this space (the double coset space) is described by a split Cartan subalgebra according to the Cartan decomposition theory. This decomposition leads to an integral formula for the invariant measure on the symmetric space involving the Haar measure on the maximal compact subgroup and the Lebesgue measure on the split Cartan subalgebra. In this talk, we study the invariant measures on a wider class of spaces, namely, reductive real spherical homogeneous spaces, which including semisimple symmetric spaces. We specifically detail the construction of Cartan decompositions and that of the corresponding integral formulas on non-symmetric reductive real spherical homogeneous spaces, providing concrete examples.

Hiroyuki Tasaki: "Polars of $Pin^c$ groups and related compact Lie groups"
We present results on polars of $Pin^c$ groups and related compact Lie groups, obtained by using maximal tori of connected compact Lie groups and their analogues in disconnected compact Lie groups. We show that every oriented real Grassmann manifolds appear as polars of $Pin^c$ groups. We also clarify the connection between polars of $Ss^c$ groups and two connected components of the orthogonal complex structures.

Daisuke Tarama: "Subriemannian geodesic flows of the seven-dimensional sphere"
This talk deals with the geodesic flows of the seven-dimensional sphere with respect to four specific trivializable subriemannian structures. The complete integrability is proved for each of the Hamiltonian flows, on the basis of a method introduced by A. Thimm at the beginning of 1980's. The subriemannian structures are described in terms of Clifford modules and the key of the proof for the complete integrability lies in construction of a chain of non-degenerate subalgebras in the Lie algebra $\mathfrak{so}(8)$. Further, the invariant differential operators corresponding to the constructed first integrals are also discussed. The talk is based on a joint work with Wolfram Bauer and Abdellah Laaroussi.

Yasuyuki Nagatomo: "The categories of representations and holomorphic maps into Grassmannians"
We will show that the category of the unitary representations of a compact, connected, simply-connected, semi-simple Lie group is isomorphic to one of full holomorphic maps of a flag manifold to complex Grassmannians with gauge condition for semi-positive Hermitian holomorphic homogeneous vector bundles.

Kurando Baba: "On Backward Mean Curvature Flow for Equifocal Submanifolds in Symmetric Spaces of Compact Type"
It is known that the backward mean curvature flows starting from equifocal submanifolds have long-time solutions and converge to minimal equifocal submanifolds. In this talk, we give observations of these long-time solutions in terms of the tangential focal datum associated with the initial equifocal submanifolds. This talk is based on joint work with Naoyuki Koike.

Masahiro Morimoto: "Affine differential geometry of parallel transport maps and weakly reflective submanifolds"
In the 1990s, C.-L. Terng and G. Thorbergsson investigated a natural Riemannian submersion from an infinite dimensional Hilbert space onto a compact Riemannian symmetric space G/K. This map is called the parallel transport map over G/K. Afterward, N. Koike extended their theory to the case that G/K is a Riemannian symmetric space of non-compact type. In this talk, I will explain my research on the extension of those theories in the framework of affine differential geometry. In particular, the parallel transport map over an affine symmetric space is defined and shown to be an affine submersion with horizontal distribution in the sense of Abe and Hasegawa. Its relation to weakly reflective submanifolds in symmetric spaces will also be discussed.

Yuta Yamauchi: "The total absolute curvature of submanifolds with singularities"
For an $n$-dimensional immersed compact submanifold in Euclidean space $\mathbb{R}^{n+r}$, it is known that the total absolute curvature is greater than or equal to the sum of the Betti numbers. Moreover, the total absolute curvature is equal to $2$ if and only if the submanifold is a convex hypersurface embedded in an affine $(n+1)$-subspace of $\mathbb{R}^{n+r}$ (the Chern-Lashof theorem). In this talk, we show a Chern-Lashof type theorem for submanifolds with singularities in Euclidean space. More precisely, we prove that for an $n$-dimensional admissible compact frontal in $\mathbb{R}^{n+r}$, its total absolute curvature is greater than or equal to the sum of the Betti numbers. Furthermore, if the total absolute curvature is equal to $2$ and all singularities are of the first kind, then the image of the frontal coincides with a closed convex domain of an affine $n$-subspace of $\mathbb{R}^{n+r}$.

Date: 9-10, December, 2025