  We obtain some new results on classical solutions of two dimensional
Euclidean sigma models. From earlier work of Din-Zakrzewski, Glaser-Stora, and
numerous differential geometers, one knows explicit solutions in the case of
the $S^n$-model, the $CP^n$-model, and the $U(n)$-model. However, very little
is known about the "moduli space" of solutions itself. In this paper we study
the connected components of these spaces. In a subsequent paper (with M. Furuta
and M. Kotani), we compute the fundamental group, in the case of the
$S^n$-model.
