Length spectrum of complete simply connected Sasakian space forms

Abstract We investigate the distribution of lengths of closed geodesics on complete and simply connected Sasakian space forms of dimension greater than 1. When its constant ϕ-sectional curvature k is irrational and satisfies k > − 9 / 4 or k − 4 , we can see that two closed geodesics on this manifold are congruent to each other in strong sense by an isometry of this manifold if and only if they have a common length. When k is rational and k ≠ − 3 , 1 , the number of congruence classes of closed geodesics of given length is finite, but not uniformly finite, and its growth order with respect to their lengths is less than polynomial order.