Makiko Sumi Tanaka

Antipodal sets of symmetric

We show that antipodal sets of symmetric R-spaces have the following properties. Any antipodal set is included in a great antipodal set and any two great antipodal sets are congruent.

R

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.

-spaces

We give a different proof of a theorem of O. Loos [5] which char acterizes maximal tori of extrinsically symmetric spaces. On the way we sh ow some facts on certain symmetric subspaces, so called meridians, which previ ously have been known only using classification.

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 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.

MAXIMAL TORI OF EXTRINSIC SYMMETRIC SPACES AND MERIDIANS

We classify maximal antipodal subgroups of the group \(\mathrm {Aut}(\mathfrak {g})\) of automorphisms of a compact classical Lie algebra \(\mathfrak {g}\). A maximal antipodal subgroup of \(\mathrm {Aut}(\mathfrak {g})\) gives us as many mutually commutative involutions of \(\mathfrak {g}\) as possible. For the classification we use our former results of the classification of maximal antipodal subgroups of quotient groups of compact classical Lie groups. We also use canonical forms of elements in a compact Lie group which is not connected.

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

We look back at the history of symmetric R-spaces and give a survey of the geometry of symmetric R-spaces including the author’s recent results.