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

2013/02/12 by Zizhou Tang, Tang, Zizhou, Weiping Zhang +1
Mathematics · Physics and Astronomy · #53C #58J #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Mathematics and Applications #math.DG #math.GT #msc:53C #msc:58J

paper · pdf · doi:10.48550/arxiv.1302.2792

7 pages, final version to appear in Advances in Mathematics

openalex publication_date 2013/02/12 · arxiv created 2013/12/22 · arxiv updated 2013/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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