Can a Lie group be of negative sectional curvature? In fact every Lie group contains an entire curve which means that a Lie group does not admit a complete metric with negative sectional curvature
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4$\begingroup$ This question makes sense only if you say more precisely how you define a metric on the Lie group: what kind of metric, satisfying which conditions? $\endgroup$– YCorCommented Jan 1, 2020 at 16:40
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$\begingroup$ I'm talking about the Hermitian metric (defined by a positive (1,1)form ) $\endgroup$– SamirCommented Jan 1, 2020 at 18:09
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2$\begingroup$ I agree that this is still unclear — but would guess that Milnor’s “Special Example 1.7” gives a positive ($=$ negatively curved...) answer to what is probably intended. Does it? $\endgroup$– Francois ZieglerCommented Jan 2, 2020 at 7:29
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$\begingroup$ The problem for me is: how a complex Lie group on the one hand admits a Riemannian metric with negative sectional curvature and on the other hand it contains copies of \ C therefore it contains entire curves (image of \ C by holomorphic function ) and we know that the presence of an entire curve implies the inexistence of a metric with negative sectional curvature $\endgroup$– SamirCommented Jan 2, 2020 at 18:01
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