Using the theory of normal cycles, we associate with each geometric subset of a Riemannian manifold a —tensor-valued— curvature measure, which we call its second fundamental measure. This measure provides a finer description of the geometry of singular sets than the standard curvature measures. Moreover, we deal with approximation of curvature measures. We get a local quantitative estimate of the difference between curvature measures of two geometric subsets, when one of them is a smooth hypersurface.
"Second fundamental measure of geometric sets and local approximation of curvatures." J. Differential Geom. 74 (3) 363 - 394, November 2006. https://doi.org/10.4310/jdg/1175266231