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Question : I have a question about the notion of hyperbolicity in the sense of Koszul.

Koszul asserts that locally flat compact hyperbolic manifolds do not contain a parallel vector field. Do you have any idea of the proof? I'm curious to see the idea of proof.

Thank you.

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  • $\begingroup$ Definition of hyperbolicity? Normally that and locally flat contradict each other. $\endgroup$
    – Deane Yang
    Commented Feb 22 at 19:30
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    $\begingroup$ @DeaneYang: It is in the sense of flat affine structures, cf. mathoverflow.net/questions/455266/… $\endgroup$
    – Moishe Kohan
    Commented Feb 22 at 19:32
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    $\begingroup$ I think integrating a parallel vector field yields a complete affine geodesic contradicting hyperbolicity. $\endgroup$
    – Moishe Kohan
    Commented Feb 22 at 20:51
  • $\begingroup$ @MoisheKohan, thanks. What are fundamental examples? $\endgroup$
    – Deane Yang
    Commented Feb 23 at 1:13
  • $\begingroup$ @DeaneYang: A compact hyperbolic manifold times circle. The developing map is onto the future light cone. Similarly, examples coming from cocompact lattices in $GL(n, R)$. Developing maps are onto the cone of positive definite matrices. $\endgroup$
    – Moishe Kohan
    Commented Feb 23 at 1:39

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