Projective and conformal closed manifolds with a higher-rank lattice action

TL;DR

This paper proves that closed manifolds with certain geometric structures acted upon by higher-rank lattices are globally equivalent to model spaces or their covers, extending understanding of geometric actions in higher-rank Lie group contexts.

Contribution

It establishes the global equivalence of manifolds with projective or conformal structures under higher-rank lattice actions to model spaces or their covers, especially in the maximal real-rank case.

Findings

Manifolds are globally equivalent to model spaces or their double covers in maximal real-rank cases.

The results unify projective and conformal geometry contexts through parallel proofs.

The paper extends previous local results to global classifications for these geometric structures.

Abstract

We prove global results about actions of cocompact lattices in higher-rank simple Lie groups on closed manifolds endowed with either a projective class of connections or a conformal class of pseudo-Riemannian metrics of signature $(p,q)$, with $\min(p,q) \geq 2$. In the continuity of a recent article, provided that such a structure is locally equivalent to its model $\mathbf{X}$, the main question treated here is the completeness of the associated $(G,\mathbf{X})$-structure. Because of the similarities between the model spaces of projective geometry and non-Lorentzian conformal geometry, a number of arguments apply in both contexts. We therefore present the proofs in parallel. The conclusion is that in both cases, when the real-rank is maximal, the manifold is globally equivalent to either the model space $\mathbf{X}$ or its double cover.