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Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
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Curvature $\geq-1$ but not $\geq1$
$\geq k$, but not of curvature $\geq k+\varepsilon$ for all $\varepsilon>0$. … Note that the curvature bound is from below.
I apologize to the experts if the terminology is not standard. …