Hiroyuki Tasaki

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 reduce the problem of classifying all maximal antipodal sets in the oriented real Grassmann manifold to that of classifying all maximal subsets satisfying certain conditions in the set consisting of subsets of cardinality k in {1, …, n}. Using this reduction we classify all maximal antipodal sets in for k ≤ 4. We construct some maximal antipodal subsets for higher k.

-spaces

We represent the convergence rates of the Riemann sums and the trapezoidal sums with respect to regular divisions and optimal divisions of a bounded closed interval to the Riemann integrals as some limits of their expanded error terms.

ANTIPODAL SETS IN ORIENTED REAL GRASSMANN MANIFOLDS

We give a Poincaré formula for any real surfaces in the complex projective plane which states that the mean value of the intersection numbers of two real surfaces is equal to the integral of some terms of their Kähler angles.

Convergence rates of approximate sums of Riemann integrals

Abstract Let M be a compact symmetric space, and K the isotropy subgroup of the group of all isometries of M at a point o of M . We consider two actions of K , namely the natural action of K on M and the linear isotropy action of K on the tangent space T o M . In both cases, we show that in each category of orbits of the “same type” under K there exists a unique minimal one.

Integral Geometry of Real Surfaces in the Complex Projective Plane

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.

Minimal orbits of the isotropy groups of symmetric spaces of compact type

We estimate the cardinalities of antipodal sets in oriented real Grassmann manifolds of low ranks. The author reduced the classification of antipodal sets in oriented real Grassmann manifolds to a certain combinatorial problem in a previous paper. So we can reduce estimates of the antipodal sets to those of certain combinatorial objects. The sequences of antipodal sets we obtained in previous papers show that the estimates we obtained in this paper are the best.

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.

Estimates of antipodal sets in oriented real Grassmann manifolds

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

In this article, we first describe antipodal sets and the structure of intersections of two real forms in complex flag manifolds. In particular, in the complex flag manifold consisting of sequences of complex subspaces in a complex vector space we investigate the real form consisting of sequences of quaternionic subspaces. Moreover, we discuss applications to the Hamiltonian volume minimizing problem.