Arithmetic distance on compact symmetric spaces

Abstract

Let M be a compact Riemannian symmetric space. Then M=G/K, where G is the identity component of the isometry group of M and K is the isotropy subgroup of G at a point. In 1965 Nagano studied and classified the geometric transformation groups of compact symmetric spaces. Roughly speaking they are ‘larger’ groups L that act on M, (i) G/L; (ii) L is a Lie transformation group acting effectively on M; (iii) L preserves the symmetric structure of M; and (iv) L is simple. Using ‘Helgason spheres’, S(α), the minimal totally geodesic spheres in a compact irreducible symmetric space, we define an arithmetic distance for compact irreducible symmetric spaces and prove: THEOREM. Let M=G p(K n ), K=ℂ, H, or R, or M=AI(n), of rank greater that 1 and dimension greater that 3, let L′ be the geometric transformation group of M. Let L={ϕ: M→M: ϕ is a diffeomorphism and ϕ preserves arithmetic distance}. Then L=L′