A fundamental differential system of Riemannian geometry

TL;DR

This paper introduces a fundamental exterior differential system for Riemannian geometry, based on the tangent sphere bundle, providing an intrinsic, invariant framework with new applications and examples.

Contribution

It develops a global differential form system for Riemannian manifolds, extending hypersurface theory and offering new tools for geometric analysis.

Findings

Established an invariant differential system of degree n for Riemannian manifolds.

Extended Euclidean hypersurface results to Riemannian settings.

Provided new applications and examples of Euler-Lagrange systems in geometry.

Abstract

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.