Geometry of curves in parabolic homogeneous spaces

TL;DR

This paper investigates the geometry of integral curves in parabolic homogeneous spaces, constructing a canonical frame and criteria for Cartan connections, with extensions to more general curves and geometries.

Contribution

It introduces a canonical moving frame for curves of constant type and provides conditions for it to form a Cartan connection, extending to broader geometric contexts.

Findings

Constructed a canonical moving frame for such curves.

Derived criteria for the moving frame to be a Cartan connection.

Discussed generalizations to higher-dimensional submanifolds and other geometries.

Abstract

The current paper is devoted to the study of integral curves of constant type in parabolic homogeneous spaces. We construct a canonical moving frame bundle for such curves and give the criterium when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and to general parabolic geometries are discussed.