Inequivalence of the various notions of completeness for Cartan geometries

Abstract

We demonstrate that the three prevailing notions of completeness for Cartan geometries are not actually equivalent and provide methods for constructing many examples in the affine case, modernizing and extending results due to Yeaton Clifton. In an effort to rectify the folklore on the topic, we also provide detailed explanations of how specious proofs in the literature that would claim that these types of completeness are equivalent break down.