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A study on maximal antipodal sets of compact symmetric spaces

佐々木, 優 筑波大学 DOI:10.15068/0002000939

2021.08.03

概要

Let M be a compact Riemannian symmetric space and denote the geodesic symmetry at x ∈ M by sx. If sx(y) = y for two points x, y ∈ M , we say that x, y are antipodal. A subset S of M is an antipodal set, if any two points of S are antipodal. The 2-number #2M of M is the maximum of cardinalities of antipodal sets of M . We call an antipodal set S of M great if #S = #2M . An antipodal set S is called maximal if there are no anitipodal sets including S properly. These notions were introduced by Chen-Nagano [2]. In general, any antipodal set of any Riemannian symmetric space of noncompact type is a one-point set, so we consider only compact symmetric spaces in this paper.

By the definition of antipodal sets, great antipodal sets are maximal. However, maximal antipodal sets are not necessarily great. It is known that any antipodal set is a finite set and #2M is finite for any compact symmetric space M . Moreover, it is known that the 2-number is an inavriant for compact symmetric spaces.

For any two compact symmetric spaces M, N , if #2M ̸= #2N , then M and N are not isomorphic. Also, it is known that the 2-number is related to the topology. Let M be a symmetric R space. Then, the followings are true [15]:

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