  We use the $\eta$-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 B\'erard-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.
