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There is a notion of combinatorial curvature due to Forman, see here (published paper) or here (preprint). I checked for a couple of small triangulations of $\mathbb{RP}^2$ (6-vertex, 7-vertex, 9-vertex) and the 1-curvature seemed to be non-positive for any edge in these. Is there any triangulation which has at least some positive curvature? (My expectation is no: subdivisions seem to have more and more negative combinatorial curvature, so positive curvature triangulations should be small.) If there is no triangulation of $\mathbb{RP}^2$ with some positive curvature, what would be the explanation?

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Take any triangulation and any pair of adjacent faces $ABC$ and $BCD$. Now subdivide the shared edge $BC$ and both faces into three edges $BE$, $EF$, $FC$ and six faces $ABE$, $AEF$, $AFC$, $BED$, $EFD$, $FCD$. The edge $EF$ has $1$-curvature $2+2-2=2$.

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  • $\begingroup$ Thanks. So does this mean that positive combinatorial curvature on $\mathbb{RP}^2$ detects possibilities to simplify the triangulation (by reverting the operation you described)? $\endgroup$ – Matthias Wendt Apr 11 '18 at 9:04

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