  We consider a nonlinear sigma model coupled to the metric of a conic space.
We obtain restrictions for a nonlinear sigma model to be a source of the conic
space. We then study nonlinear sigma model in the conic space background. We
find coordinate transformations which reduce the chiral fields equations in the
conic space background to field equations in Minkowski spacetime. This enables
us to apply the same methods for obtaining exact solutions in Minkowski
spacetime to the case of a conic spacetime. In the case the solutions depend on
two spatial coordinates we employ Ivanov's geometrical ansatz. We give a
general analysis and also present classes of solutions in which there is
dependence on three and four coordinates. We discuss with special attention the
intermediate instanton and meron solutions and their analogous in the conic
space. We find differences in the total actions and topological charges of
these solutions and discuss the role of the deficit angle.
