η-Invariant and a problem of Bérard-Bergery on the existence of closed geodesics

We use the η-invariant of Atiyah–Patodi–Singer to compute the Eells–Kuiper invariant for the Eells–Kuiper quaternionic projective plane. By combining with a known result of Bérard-Bergery, it shows that every Eells–Kuiper quaternionic projective plane carries a Riemannian metric such that all geodesics passing through a certain point are simply closed and of the same length.