Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Holonomy of parabolic geometries near isolated higher-order fixed points

Jacob W. Erickson

TL;DR

The work studies semi-global consequences of automorphisms with higher-order fixed points in parabolic Cartan geometries, proving that a dynamical isotropy on the model induces an invariant open region with trivial holonomy. It develops a holonomy toolkit—convergence of developments and antidevelopments—and an ensnaring criterion for isotropy that guarantees curvature vanishes and holonomy becomes trivial on a region near the fixed point. Using these tools, it derives explicit semi-global embeddings into flat model geometries and fully characterizes almost $c$-projective, almost quaternionic, and nondegenerate partially integrable almost CR structures admitting a higher-order fixed point. These results extend Ferrand–Obata-type rigidity to semi-global settings without requiring curvature regularity, thus broadening the scope of global conclusions drawn from automorphism behavior.

Abstract

For Cartan geometries admitting automorphisms with isotropies satisfying a particular, loosely dynamical property on their model geometries, we demonstrate the existence of an open subset of the geometry with trivial holonomy. This property, which generalizes characteristics of isotropies corresponding to isolated higher-order fixed points in parabolic geometries that are known to require a nearby open subset to have vanishing curvature, only relies upon the behavior of the isotropy in the model geometry, and therefore applies regardless of initial curvature assumptions, such as regularity or normality. Along the way to proving our main results, we also derive a couple of results for working with holonomy, relating to limits of sequences of developments and the existence of antidevelopments, that are useful in their own right. To showcase the effectiveness of the techniques developed, we use them to completely characterize all almost c-projective and almost quaternionic structures that admit a nontrivial automorphism with a higher-order fixed point, as well as all nondegenerate partially integrable almost CR structures that admit a higher-order fixed point with non-null isotropy.

Holonomy of parabolic geometries near isolated higher-order fixed points

TL;DR

The work studies semi-global consequences of automorphisms with higher-order fixed points in parabolic Cartan geometries, proving that a dynamical isotropy on the model induces an invariant open region with trivial holonomy. It develops a holonomy toolkit—convergence of developments and antidevelopments—and an ensnaring criterion for isotropy that guarantees curvature vanishes and holonomy becomes trivial on a region near the fixed point. Using these tools, it derives explicit semi-global embeddings into flat model geometries and fully characterizes almost -projective, almost quaternionic, and nondegenerate partially integrable almost CR structures admitting a higher-order fixed point. These results extend Ferrand–Obata-type rigidity to semi-global settings without requiring curvature regularity, thus broadening the scope of global conclusions drawn from automorphism behavior.

Abstract

For Cartan geometries admitting automorphisms with isotropies satisfying a particular, loosely dynamical property on their model geometries, we demonstrate the existence of an open subset of the geometry with trivial holonomy. This property, which generalizes characteristics of isotropies corresponding to isolated higher-order fixed points in parabolic geometries that are known to require a nearby open subset to have vanishing curvature, only relies upon the behavior of the isotropy in the model geometry, and therefore applies regardless of initial curvature assumptions, such as regularity or normality. Along the way to proving our main results, we also derive a couple of results for working with holonomy, relating to limits of sequences of developments and the existence of antidevelopments, that are useful in their own right. To showcase the effectiveness of the techniques developed, we use them to completely characterize all almost c-projective and almost quaternionic structures that admit a nontrivial automorphism with a higher-order fixed point, as well as all nondegenerate partially integrable almost CR structures that admit a higher-order fixed point with non-null isotropy.
Paper Structure (16 sections, 14 theorems, 78 equations, 4 figures)

This paper contains 16 sections, 14 theorems, 78 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relevant model geometries
  4. Generalities for Cartan geometries
  5. Development and holonomy
  6. Some useful tools for working with holonomy
  7. Convergence of developments
  8. Existence of antidevelopments
  9. Isotropies that "ensnare" a region of the model
  10. Criteria for the isotropy
  11. Existence of ensnared region with trivial holonomy
  12. Applications
  13. General result
  14. Proof of Theorem \ref{['thmA']}
  15. Proof of Theorem \ref{['thmB']}
  16. ...and 1 more sections

Key Result

Theorem A

Suppose $(M,J,[\nabla])$ is an almost c-projective structure. If $(M,J,[\nabla])$ admits a nontrivial automorphism $\alpha$ with a higher-order fixed point $x\in M$, so that $\alpha(x)=x$ and $\alpha_{*x}=\mathrm{id}_{T_xM}$, then $(M,J,[\nabla])$ embeds geometrically onto a dense open subspace of t

Figures (4)

  • Figure 1: Sketches of automorphisms with an isolated higher-order fixed point for 2-dimensional real projective geometry $(\mathop{\mathrm{\mathrm{PGL}}}\nolimits_3\mathbb{R},P)$ (left) and 1-dimensional complex projective geometry $(\mathop{\mathrm{\mathrm{PGL}}}\nolimits_2\mathbb{C},P)$ (right)
  • Figure 2: A concatenation of the form $\zeta_U\star q_{{}_H}(\delta)$ (left) and some of its iterates under $a$ (right)
  • Figure 3: The path $\gamma_G$ in $G$, with neighborhoods of its intersection with $G\setminus\mathscr{F}$ highlighted in gray
  • Figure 4: A path $c_\varepsilon$ in $G$, which satisfies $c_\varepsilon([0,1])\subseteq\mathscr{F}$ and coincides with $\gamma_G=c_0$ outside of the highlighted neighborhoods of the intersections of $\gamma_G$ with $G\setminus\mathscr{F}$

Theorems & Definitions (38)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 28 more