  We examine the difference between several notions of curvature homogeneity
and show that the notions introduced by Kowalski and Vanzurova are genuine
generalizations of the ordinary notion of k-curvature homogeneity. The
homothety group plays an essential role in the analysis. We give a complete
classification of homothety homogeneous manifolds which are not homogeneous and
which are not VSI and show that such manifolds are cohomogeneity one. We also
give a complete description of the local geometry if the homothety character
defines a split extension.
