Geometry and topology of maximal antipodal sets and related topics

Maximal antipodal sets of Riemannian manifolds were introduced by the author and T. Nagano in [Un invariant géométrique riemannien, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 5, 389--391]. Since then maximal antipodal sets have been studied by many mathematicians and they shown that maximal antipodal sets are related to several important areas in mathematics. The main purpose of this paper is thus to present a comprehensive survey on geometry and topology of maximal antipodal sets and also on their applications to several related topics.