A method for determining Cartan geometries from the local behavior of automorphisms

TL;DR

This work constructs a universal object, the sprawl, that encodes the local behavior of an automorphism of a Cartan geometry into a new geometry and proves a universal property ensuring a natural map from the sprawl into any geometry sharing the same local dynamics. The method hinges on increment-based path incrementation and thin-homotopy–guided gluing to produce a Hausdorff principal bundle with a sprawl map into the original geometry, yielding an automorphism that matches the local behavior. The authors then apply this framework to parabolic geometries—real projective, conformal Lorentzian, and path geometries—to build non-flat global examples with higher-order fixed points, by relating local data to the model geometry and leveraging curvature vanishing near fixed points. The approach provides a systematic way to realize and compare Cartan geometries from prescribed local symmetries, enabling explicit global constructions and embeddings into model pieces across multiple parabolic settings.

Abstract

We introduce a construction for a Cartan geometry that captures the local behavior of a given geometric automorphism near a distinguished element. The result of this construction, which we call the sprawl generated by the automorphism, is uniquely characterized by a kind of universal property that allows us to compare different Cartan geometries that admit automorphisms with equivalent local behavior near a distinguished element. As example applications, we describe how to construct non-flat real projective structures admitting nontrivial automorphisms with higher-order fixed points and extend some known local automorphisms with higher-order fixed points on non-flat parabolic geometries to global automorphisms.

Figures (6)

• Figure 1: The region $U$ (highlighted in darker gray) is pushed back to itself in the torus by iterates of the translation $\alpha$, but lifting the situation to the plane gives a situation with identical local behavior such that $U$ never returns to itself after leaving
• Figure 2: The region $U$ (highlighted in lighter gray) given by the union of an open ball and an open sector that is disjoint from its image under the rotation $\alpha$, as well as a depiction of its intersection (highlighted in darker gray) with an iterate under $\alpha$ where the overlap escapes the open ball
• Figure 3: A path $\gamma$, highlighted in darker gray, in the manifold $M$, as well the region $U$, highlighted in lighter gray, and its iterates under an automorphism $\alpha$
• Figure 4: An incrementation for the path $\gamma$ depicted in Figure \ref{['increment1pic']}
• Figure 5: The region $U=D\cup D'\cup V$ given by the union of two open disks $D$ and $D'$, one around each fixed point of a rotation on the 2-sphere, with the interior $V$ of a spherical lune connecting them

Theorems & Definitions (31)

• Definition 2.1
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