A fundamental differential system of Riemannian geometry

We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree n associated to any given oriented Riemannian manifold M of dimension n + 1. The framework is that of the tangent sphere bundle of M. We generalise to a Riemannian setting some results from the theory of hypersurfaces in flat Euclidean space. We give new applications and examples of the associated Euler-Lagrange differential systems.