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Symmetric spaces of higher rank do not admit differentiable compactifications

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Abstract

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank 1 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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Authors and Affiliations

  1. UMPA, École normale supérieure de Lyon, CNRS UMR 5669, Lyon Cedex, France

    Benoît Kloeckner

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  1. Benoît Kloeckner
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Correspondence to Benoît Kloeckner.

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Kloeckner, B. Symmetric spaces of higher rank do not admit differentiable compactifications. Math. Ann. 347, 951–961 (2010). https://doi.org/10.1007/s00208-009-0464-z

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  • DOI: https://doi.org/10.1007/s00208-009-0464-z

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