  We present a short-time existence theorem of solutions to the initial value
problem for Schroedinger maps of a closed Riemannian manifold to a compact
almost Hermitian manifold. The classical energy method cannot work for this
problem since the almost complex structure of the target manifold is not
supposed to be parallel with respect to the Levi-Civita connection. In other
words, a loss of one derivative arises from the covariant derivative of the
almost complex structure. To overcome this difficulty, we introduce a bounded
pseudodifferential operator acting on sections of the pullback bundle, and
essentially eliminate the loss of one derivative from the partial differential
equation of the Schroedinger map.

Analysis of PDEs;Differential Geometry