  We introduce a class of maps from an affine flat into a Riemannian manifold
that solve an elliptic system defined by the natural second order elliptic
operator of the affine structure and the nonlinear Riemann geometry of the
target. These maps are called affine harmonic. We show an existence result for
affine harmonic maps in a given homotopy class when the target has non positive
sectional curvature and some global non triviality condition is met. An example
shows that such a condition is necessary.
  The analytical part is made difficult by the absence of a variational
structure underlying affine harmonic maps. We therefore need to combine
estimation techniques from geometric analysis and PDE theory with global
geometric considerations.
