$\eta$-invariant and a problem of B\'erard-Bergery on the existence of closed geodesics

TL;DR

This paper employs the $\eta$-invariant to compute the Eells-Kuiper invariant for a special quaternionic projective plane, demonstrating the existence of metrics with all geodesics through a point being closed and equal in length.

Contribution

It introduces a novel use of the $\eta$-invariant to analyze geodesic properties on quaternionic projective planes, linking topological invariants with geometric structures.

Findings

Every Eells-Kuiper quaternionic projective plane admits a metric with all geodesics through a point being closed and of the same length.

The $\eta$-invariant effectively computes the Eells-Kuiper invariant in this context.

The results connect topological invariants with geometric properties of specific manifolds.

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 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.