Symmetric spaces of higher rank do not admit differentiable compactifications

Details Symmetric spaces of higher rank do not admit differentiable compactifications Symmetric spaces of higher rank do not admit differentiable compactifications

2009-12-04

arxiv.org/abs/0912.0814v1

Mathematische Annalen 347, 4 (2010) 951-961

Mathematics

Differential Geometry

13 pages, to appear in Mathematische Annalen

Scientific paper

10.1007/s00208-009-0464-z

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.

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