Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces

Abstract

The Borsuk-Ulam theorem is a well-known theorem in algebraic topology which states that if φ : S n → R k is a continuous map from the unit n-sphere into the Euclidean k-space with k ≤ n, then there is a pair of antipodal points on S n that are mapped by φ to the same point in R k .In this paper we study continuous real-valued functions of compact symmetric spaces.The main result states that if f : M → R is a continuous isotropic function of a compact symmetric space M into the real line R. Then f carries some maximal antipodal set of M to the same point in R, whenever M is one of the following spaces: Spheres; the projective spaces FP n (F = R, C, H); the Cayley plane F II; the exceptional spaces EIV ; EIV * ; GI; and the exceptional Lie group G 2 .Some additional related results are also given.