Isometry types of frame bundles

TL;DR

This paper characterizes when the orthonormal frame bundles of two closed Riemannian manifolds are isometric, showing they are if and only if the manifolds are, with exceptions in dimensions 3, 4, and 8.

Contribution

It proves a near-complete characterization of isometry types of frame bundles in relation to the base manifolds, addressing a question posed by Benson Farb.

Findings

Frame bundles are isometric if and only if the base manifolds are isometric, except possibly in dimensions 3, 4, and 8.

The result provides a new understanding of the relationship between manifold geometry and their frame bundles.

The paper resolves a previously open question in Riemannian geometry about the isometry types of frame bundles.

Abstract

We consider the orthonormal frame bundle F(M) of a Riemannian manifold M. A construction of Sasaki defines a canonical Riemannian metric on F(M). We prove that for two closed Riemannian n-manifolds M and N, the frame bundles F(M) and F(N) 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.