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In this paper the arising of Gribov copies both in Landau and Coulomb gaugesin regions with non-trivial topologies but flat metric, (such as closed tubesS1XD2, or RXT2) will be analyzed. Using a novel generalization of the hedgehogansatz beyond spherical symmetry, analytic examples of Gribov copies of thevacuum will be constructed. Using such ansatz, we will also construct theelliptic Gribov pendulum. The requirement of absence of Gribov copies of thevacuum satisfying the strong boundary conditions implies geometricalconstraints on the shapes and sizes of the regions with non-trivial topologies.

Generalized Hedgehog ansatz and Gribov copies in regions with non trivial topologies