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Small covers were introduced by Davis and Januszkiewicz in 1991. We introducethe notion of equilibrium triangulations for small covers. We study equilibriumand vertex minimal $\mathbb{Z}_2^2$-equivariant triangulations of$2$-dimensional small covers. We discuss vertex minimal equilibriumtriangulations of $\mathbb{RP}^3 \# \mathbb{RP}^3$, $S^1 \times \mathbb{RP}^2$and a nontrivial $S^1$ bundle over $\mathbb{RP}^2$. We construct some niceequilibrium triangulations of the real projective space $\mathbb{RP}^n$ with$2^n +n+1$ vertices. The main tool is the theory of small covers.

Equilibrium and equivariant triangulations of some small covers with minimum number of vertices