The number of geometrically distinct reversible closed geodesics on a Finsler sphere with K≡ 1

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In this paper we study the Finsler sphere (Sⁿ,F) with n>1, which has constant flag curvature K≡ 1 and only finite prime closed geodesics. In this case, the connected isometry group I₀(Sⁿ,F) must be a torus which dimension satisfies 0< I(Sⁿ,F) ≤[n+1/2]. We will prove that the number of geometrically distinct reversible closed geodesics on (Sⁿ,F) is at least I(Sⁿ,F). When I₀(Sⁿ,F)=[n+1/2], the equality happens, and there are exactly 2[n+1/2] prime closed geodesics, which verifies Anosov conjecture in this special case.