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When is a simply connected Lie group with an invariant metric of positive sectional curvature compact?

The point of the question is: does this hypothesis provide a positive lower bound on Ricci curvature in the sense of Bonnet-Meyers's theorem on the metric of positive sectional curvature? If not, under what conditions is it possible? Is there any counterexample? If Bonnet-Meyers is not satisfied, is the group still compact? I mean, are there any obstructions in terms of the Lie algebra of the group?

Thanks

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    $\begingroup$ Positive sectional curvature implies positive Ricci curvature, so you're done by Bonnet--Myers (and simple connectivity is irrelevant). But you've included this argument in the question statement, so what are you asking? $\endgroup$
    – John Pardon
    Commented Aug 30, 2017 at 0:29
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    $\begingroup$ In particular the lie group structure means that positive sectional curvature implies strictly positive sectional curvature. $\endgroup$
    – Tim Carson
    Commented Aug 30, 2017 at 0:32
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    $\begingroup$ In particular, the Ricci curvature is positive at each point. You're worried it might not be uniformly bounded away from 0? But by the invariance of the metric, the Ricci curvature is "constant", i.e. if $\mathrm{Ric} \ge k > 0$ at one point, then the same is true everywhere. $\endgroup$
    – Nate Eldredge
    Commented Aug 30, 2017 at 0:32
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    $\begingroup$ The answer I was expecting was given by @NateEldredge, I was not sure about it and as you claimed it trivially follows. Thanks. $\endgroup$
    – L.F. Cavenaghi
    Commented Aug 30, 2017 at 0:57
  • $\begingroup$ It could happen the manifold be like a paraboloid that is not compact and the curvature even positive approaches widely of zero. $\endgroup$
    – L.F. Cavenaghi
    Commented Aug 30, 2017 at 0:58

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