Prime geodesic periods on manifolds of complex projective type

Let M²ⁿ, n ≥ 2, be a closed connected Riemannian manifold all of whose geodesics are closed, and assume H²(M; ℚ) ≠ 0. We prove that all prime unit-speed geodesics have the same period for 2 ≤ n ≤ 8. For every n ≥ 2, the same conclusion holds if each nontrivial stabilizer of the geodesic circle action has prime-power order. For n ≥ 3, unequal prime periods imply the existence of a proper connected critical manifold whose generic stabilizer order has at least two distinct odd prime divisors and whose inverse image in an auxiliary circle bundle has total rational Betti number at least eight. The period restrictions follow from reversal in equivariant Morse theory, the cohomology of fixed manifolds of a lifted geodesic flow, and index and nullity formulas under iteration.