  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.
