Strongly essential flows on irreducible parabolic geometries

TL;DR

This paper investigates the local geometry of irreducible parabolic geometries with strongly essential flows, revealing that such flows often imply local flatness of the structure near fixed points, with new rigidity results and specific cases analyzed.

Contribution

It provides new rigidity results for irreducible parabolic geometries with strongly essential flows and demonstrates flatness near fixed points in several geometric structures.

Findings

Strongly essential flows often imply local flatness of the geometry.

New rigidity results for irreducible parabolic geometries.

Flatness established near fixed points for c-projective and quaternionic structures.

Abstract

We study the local geometry of irreducible parabolic geometries admitting strongly essential flows; these are flows by local automorphisms with higher-order fixed points. We prove several new rigidity results, and recover some old ones for projective and conformal structures, which show that in many cases the existence of a strongly essential flow implies local flatness of the geometry on an open set having the fixed point in its closure. For almost c-projective and almost quaternionic structures we can moreover show flatness of the geometry on a neighborhood of the fixed point.