= Isometry dimension of a compact flat manifold
{title2=$\dim\operatorname{Isom}_0(M)=\dim(\mathbb R^n)^{\operatorname{Hol}(M)}$}
On a compact <flat Riemannian manifold>, the integrated <Bochner identity for Killing vector fields> makes every <Killing vector field> parallel. Parallel vector fields are exactly the fixed vectors of the linear holonomy. Thus the six orientable closed flat three-manifold types have identity-component isometry dimensions three for the torus, one for each cyclic nontrivial holonomy, and zero for the <Hantzsche-Wendt manifold>.
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