Geometric Properties of Conformal Transformations on $\mathbb{R}^{p,q}$

TL;DR

This paper investigates how conformal transformations on generalized Minkowski spaces map hyperboloids and affine hyperplanes, revealing their orbit structures and extending classical properties of Möbius transformations to higher dimensions.

Contribution

It characterizes the action of conformal transformations on hyperboloids and affine hyperplanes in $\,\mathbb{R}^{p,q}$, generalizing known properties of Möbius transformations.

Findings

Transformations map hyperboloids and affine hyperplanes into similar structures.

The action is transitive when either p or q is zero.

Exactly three orbits exist when p, q ≠ 0.

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 M\"{o}bius (or fractional linear) transformations on the complex plane $\mathbb{C}$.