Symmetric spaces of higher rank do not admit differentiable compactifications

TL;DR

This paper proves that higher rank symmetric spaces cannot be equipped with differentiable compactifications, unlike rank one spaces, thus limiting the potential for new geometric structures in higher ranks.

Contribution

It establishes a negative result showing higher rank symmetric spaces do not admit differentiable compactifications, contrasting with the rank one case.

Findings

Higher rank symmetric spaces lack differentiable compactifications.

Rank one symmetric spaces can be endowed with differentiable structures at infinity.

The absence of such structures in higher ranks restricts geometric exploration.

Abstract

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or quaternionic CR depending of the space). One can ask whether such a differentiable compactification exists for higher rank spaces, hopefully leading to some knew geometry to explore. In this paper we answer negatively.