$\eta$-invariant and a problem of Bérard-Bergery on the existence of closed geodesics

Abstract

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 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.

arXiv comment7 pages, final version to appear in Advances in Mathematics