Riemannian Questions with a Fundamental Differential System

Abstract

We introduce the reader to a fundamental exterior differential system of Riemannian geometry which arises naturally with every oriented Riemannian n + 1-manifold M. Such system is related to the well-known metric almost contact structure on the unit tangent sphere bundle SM, so we endeavor to include the theory in the field of contact systems. Our EDS is already known in dimensions 2 and 3, where it was used by Griffiths in applications to mechanical problems and Lagrangian systems. It is also known in any dimension but just for flat Euclidean space. Having found the Lagrangian forms $\alpha _{i} \in \varOmega ^{n}$, 0 ≤ i ≤ n, we are led to the associated functionals $\mathcal{F}_{i}(N) =\int _{N}\alpha _{i}$, on the set of hypersurfaces N ⊂ M, and to their Poincaré-Cartan forms. A particular functional relates to scalar curvature and thus we are confronted with an interesting new equation.