In the Analysis of shape, the equivalence class of geometrical objects modulo, say, the group of similarities are modelled as elements of a shape space which usually comes with a canonical non-Euclidean Riemannian structure. E.g. filtering out size naturally leads to a sphere, filtering out rotation adds additional curvature possibly creating singularities in the shape space. While over the last decades statisticians have used Euclidean approximations to these spaces thus making tools of classical multivariate analysis available for ``sufficiently concentrated data'', we aim at intrinsic generalizations of PCA and MANOVA.