Summary: Let \(G\) be a directed graph embedded in a surface. A map \(\phi : E(G) \rightarrow \mathbb{R} \) is a tension if for every circuit \(C \subseteq G\), the sum of \(\phi\) on the forward edges of \(C\) is equal to the sum of \(\phi\) on the backward edges of \(C\). If this condition is satisfied for every circuit of \(G\) which is a contractible curve in the surface, then \(\phi\) is a local tension. If \(1 \leq|\phi(e)|\leq \alpha-1\) holds for every \(e \in E(G)\), we say that \(\phi\) is a (local) \(\alpha\)-tension. We define the circular chromatic number and the local circular chromatic number of \(G\) by \(\chi_c(G) =\inf \{\alpha \in \mathbb{R}\mid \text{}G\) has an \(\alpha\)-tension