Geometric properties of conformal transformations on $$\mathbb {R}^{p,q}$$ R p , q

Abstract

We show that conformal transformations on the generalized Minkowski space \(\mathbb {R}^{p,q}\) map hyperboloids and affine hyperplanes into hyperboloids and affine hyperplanes. We also show that this action on hyperboloids and affine hyperplanes is transitive when \(p\) or \(q\) is \(0\), and that this action has exactly three orbits if \(p, q \ne 0\). Then we extend these results to hyperboloids and affine planes of arbitrary dimension. These properties generalize the well-known properties of Möbius (or fractional linear) transformations on the complex plane \(\mathbb {C}\).