In this paper we establish the equivalence of solutions between Schr��dinger map into $\mathbb{S}^2$ or $ \mathbb{H}^2$ and their associated gauge invariant Schr��dinger equations. We also establish the existence of global weak solutions into $\mathbb{H}^2$ in two space dimensions. We extend these ideas for maps into compact hermitian symmetric manifolds with trivial first cohomology.

19 pages, submitted