The topological completion of a bilinear form

Let M = M-n,M-m be the Euclidean space R-P equipped with a symmetric bilinear form B-M of rank p = n + m and signature n - m. We compactify M so that M-c is homogeneous and has as its group of isometrics a Lie group whose dimension is the dimension of M plus 2p + 1. We observe that M-c is in two ways the total space of a non-trivial sphere bundle with base space real projective space. The compactification is well understood in the classical case when M is Minkowski space. The contribution here is to observe that the construction works generally and that it admits a natural bundle description. (C) 2002 Elsevier Science B.V. All rights reserved.