  In this paper we establish the equivalence of solutions between Schr\"odinger
map into $\mathbb{S}^2$ or $ \mathbb{H}^2$ and their associated gauge invariant
Schr\"odinger 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.
