BPS-monopoles is 3-spaces of Euclide, Riemann, Lobachevsky
Procedure of finding of the Bogomolny - Prasad - Sommerfield monopole solutions in the Georgi - Glashow model is investigated in detail on the backgrounds of three space models of constant curvature: Euclid, Riemann, Lobachevsky. Classification of possible solutions is given. It is shown that among all solutions there exist just three ones which reasonably and in a one-to-one correspondence can be associated with respective geometries. It is pointed out that the known non-singular BPS-solution in the flat Minkowski space can be understood as a result of somewhat artificial combining the Minkowski space background with a possibility naturally linked up with the Lobachewsky geometry. The standpoint is brought forth that of primary interest should be regarded only three specifically distinctive solutions - one for every curved space background. In the framework of those arguments the generally accepted status of the known monopole BPS-solution should be critically reconsidered and even might be given away.
Joint work with S.Yu. Sakovich (B.I. Stepanov Institute of Physics, Minsk, Belarus).