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Curvature $\geq-1$ but not $\geq1$
(Edited again)
In the following, for brevity, I will say that $$X\ \ \mathrm{has}\ \ \kappa_{\mathrm{max}}=k$$ if $X$ is a compact ($n$-dimensional with $n\geq2$, with empty boundary) Aleksandrov ...
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