  One of the known mathematical descriptions of singularities in General
Relativity is the b-boundary, which is a way of attaching endpoints to
inextendible endless curves in a spacetime. The b-boundary of a manifold M with
connection is constructed by forming the Cauchy completion of the frame bundle
LM equipped with a certain Riemannian metric, the b-metric G. We study the
geometry of (LM,G) as a Riemannian manifold in the case when the connection is
the Levi-Civita connection of a Lorentzian metric g on M. In particular, we
give expressions for the curvature and discuss the isometries and the geodesics
of (LM,G) in relation to the geometry of (M,g).
