  Given a compact oriented triangulated $3$-manifold we find a non-trivial
condition satisfied by certain labelings of the tetrahedra by elements of an
arbitrary abelian group which we call angle structures. Smoothness of the
manifold is used in an essential way. This is inspired by the notion of the
volume of hyperbolic manifolds, which would correspond to the case when the
abelian group is the multiplicative group of $\mathbb{C}$, but the construction
here seems to be more general, in particular it only uses the abelian group
structure.
