Cycles Cross Ratio: an Invitation

TL;DR

The paper introduces the cycles cross ratio, a M"obius invariant measure extending the classic cross ratio to spheres and conformal geometries, with potential for broad applications.

Contribution

It defines a new invariant called cycles cross ratio, generalizing the classic cross ratio to spheres and conformal geometries, and explores its fundamental properties.

Findings

Defines cycles cross ratio as a M"obius invariant measure.

Establishes the invariant's relation to inversion and anharmonicity.

Provides a framework for further exploration of properties.

Abstract

The paper introduces cycles cross ratio, which extends the classic cross ratio of four points to various settings: conformal geometry, Lie spheres geometry, etc. Just like its classic counterpart cycles cross ratio is a measure of anharmonicity between spheres with respect to inversion. It also provides a M\"obius invariant distance between spheres. Many further properties of cycles cross ratio awaiting their exploration. In abstract framework the new invariant can be considered in any projective space with a bilinear pairing.