Osamu Ikawa

The fixed point set of a holomorphic isometry, the intersection of two real forms in a Hermitian symmetric space of compact type and symmetric triads

We show a necessary and sufficient condition that the fixed point set of a holomorphic isometry and the intersection of two real forms of a Hermitian symmetric space of compact type are discrete and prove that they are antipodal sets in the cases. We also consider some relations between the intersection of two real forms and the fixed point set of a certain holomorphic isometry.

The Fixed Point Set of a Holomorphic Isometry and the Intersection of Two Real Forms in the Complex Grassmann Manifold

We study the fixed point set of a holomorphic isometry of the complex Grassmann manifold and the intersection of two real forms which are congruent to the real Grassmann manifold. Furthermore, we investigate the relation between them.

Canonical Forms Under Certain Actions on the Classical Compact Simple Lie Groups

A maximal torus of a compact connected Lie group can be seen as a canonical form of adjoint action since any two maximal tori can be transformed each other by an inner automorphism. A. Kollross defined a σ-action on a compact Lie group which is a generalization of the adjoint action. Since a σ-action is hyperpolar, it has a canonical form called a section. In this paper we study the structure of the orbit space of a σ-action and properties of each orbit, such as minimal, austere and totally geodesic, using symmetric triads introduced by the author, when σ is an involution of outer type on the compact simple Lie groups of classical type. As an application, we investigate the fixed point set of a holomorphic isometry of an irreducible Hermitian symmetric space of compact type which does not belong to the identity component of the group of holomorphic isometries.

A Duality Between Compact Symmetric Triads and Semisimple Pseudo-Riemannian Symmetric Pairs with Applications to Geometry of Hermann Type Actions

This is a survey paper of not-yet-published papers listed in the reference as [1, 2, 3]. We introduce the notion of a duality between commutative compact symmetric triads and semisimple pseudo-Riemannian symmetric pairs, which is a generalization of the duality between compact/noncompact Riemannian symmetric pairs. As its application, we give an alternative proof for Berger’s classification of semisimple pseudo-Riemannian symmetric pairs from the viewpoint of compact symmetric triads. More precisely, we give an explicit description of a one-to-one correspondence between commutative compact symmetric triads and semisimple pseudo-Riemannian symmetric pairs by using the theory of symmetric triads introduced by the second author. We also study the action of a symmetric subgroup of G on a pseudo-Riemannian symmetric space G / H, which is called a Hermann type action. For more details, see [1, 2, 3].