Isometry types of frame bundles

Wouter van Limbeek

Vol. 285 (2016), No. 2, 393–426

DOI: 10.2140/pjm.2016.285.393

Abstract

We consider the oriented orthonormal frame bundle $SO\left(M\right)$ of an oriented Riemannian manifold $M$. The Riemannian metric on $M$ induces a canonical Riemannian metric on $SO\left(M\right)$. We prove that for two closed oriented Riemannian $n$-manifolds $M$ and $N$, the frame bundles $SO\left(M\right)$ and $SO\left(N\right)$ are isometric if and only if $M$ and $N$ are isometric, except possibly in dimensions 3, 4, and 8. This answers a question of Benson Farb except in dimensions 3, 4, and 8.

Keywords

frame bundle, Sasaki–Mok metric, large group action, symmetry degree

Mathematical Subject Classification 2010

Primary: 53C24, 57S15

Secondary: 57R25, 55R10

Milestones

Received: 30 July 2014

Revised: 13 August 2016

Accepted: 26 August 2016

Published: 21 November 2016