Journal of the
Korean Mathematical Society JKMS

On $\mathbb R^{p+q+1}$ with the non-degenerate symmetric bilinear form $F$ of type $(p,q+1)$, the pseudo-sphere is defined by $S^{p,q}=\{ x\in \mathbb R^{p+q+1}:\ F(x,x)=1 \} \approx\mathbb R^p\times S^q\subset\mathbb R^{p,q+1}$. In this paper, we shall study the geometries modelled on the orthonormal frame bundle SO$(S^{p,q})$ of the pseudo-sphere $S^{p,q}$. This is a principal $G=\mbox{SO}_0(p,q)$-bundle over $S^{p,q}$. With respect to a natural pseudo-Riemannian metric, the group of weakly $G$-equivariant isometries of $\mbox{SO}(S^{p,q})$ turns out to be $\mbox{Isom}^0_G(\mbox{SO}(S^{p,q}))={\mbox{SO}_0(p,q+1)}\times {\mbox{SO}_0(p,q)}$.

Keywords: frame bundle, indefinite metric, isometry group

MSC numbers: Primary 53C50; Secondary 53C50