Abstract
For a large class of self-similar sets F in R-d, analogues of the higher-order mean curvatures of differentiable sub-manifolds 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 total curvature results.