On three recent applications of a fundamental differential system of Riemannian geometry

We recall the construction of a fundamental differential system of Riemannian geometry and survey on some of its applications. First, on the determination of the Euler characteristic of hypersurfaces in space forms. Secondly, the generalization of Hsiung-Minkowski identities concerning total mean curvatures of hypersurfaces. And thirdly, we discuss the study of the volume of unit vector fields on a Riemannian manifold via calibrations induced from the differential system.