  For a large class of self-similar sets F in R^d analogues of the higher order
mean curvatures of differentiable submanifolds are introduced, in particular,
the fractal Gauss-type curvature. They are shown to be the densities of
associated fractal curvature measures, which are all multiples of the
corresponding Hausdorff measures on F, due to its self-similarity. This local
approach based on ergodic theory for an associated dynamical system enables us
to extend former global curvature results.
