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′
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Peterson, S.P. Arithmetic distance on compact symmetric spaces. Geom Dedicata 23, 1–14 (1987). https://doi.org/10.1007/BF00147387
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DOI: https://doi.org/10.1007/BF00147387