  In contrast to graph-based models for complex networks, hypergraphs are more
general structures going beyond binary relations of graphs. For graphs,
statistics gauging different aspects of their structures have been devised and
there is undergoing research for devising them for hypergraphs. Forman-Ricci
curvature is a statistics for graphs, which is based on Riemannian geometry,
and that stresses the relational character of vertices in a network through the
analysis of edges rather than vertices. In spite of the different applications
of this curvature, it has not yet been formulated for hypergraphs. Here we
devise the Forman-Ricci curvature for directed and undirected hypergraphs,
where the curvature for graphs is a particular case. We report its upper and
lower bounds and the respective bounds for the graph case. The curvature
quantifies the trade-off between hyperedge(arc) size and the degree of
participation of hyperedge(arc) vertices in other hyperedges(arcs). We
calculated the curvature for two large networks: Wikipedia vote network and
\emph{Escherichia coli} metabolic network. In the first case the curvature is
ruled by hyperedge size, while in the second by hyperedge degree. We found that
the number of users involved in Wikipedia elections goes hand-in-hand with the
participation of experienced users. The curvature values of the metabolic
network allowed detecting redundant and bottle neck reactions. It is found that
ADP phosphorilation is the metabolic bottle neck reaction but that the reverse
reaction is not that central for the metabolism.
