We consider the oriented orthonormal frame bundle
of an oriented Riemannian
manifold
. The Riemannian
metric on
induces a canonical
Riemannian metric on
.
We prove that for two closed oriented Riemannian
-manifolds
and
, the frame
bundles
and
are isometric
if and only if
and
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