  Using Hilbert's criterion, we consider the stress-energy tensor associated to
the bienergy functional. We show that it derives from a variational problem on
metrics and exhibit the peculiarity of dimension four. First, we use this
tensor to construct new examples of biharmonic maps, then classify maps with
vanishing or parallel stress-energy tensor and Riemannian immersions whose
stress-energy tensor is proportional to the metric.
