  In this paper the arising of Gribov copies both in Landau and Coulomb gauges
in regions with non-trivial topologies but flat metric, (such as closed tubes
S1XD2, or RXT2) will be analyzed. Using a novel generalization of the hedgehog
ansatz beyond spherical symmetry, analytic examples of Gribov copies of the
vacuum will be constructed. Using such ansatz, we will also construct the
elliptic Gribov pendulum. The requirement of absence of Gribov copies of the
vacuum satisfying the strong boundary conditions implies geometrical
constraints on the shapes and sizes of the regions with non-trivial topologies.
