Special vector fields on Riemannian manifolds of constant negative sectional curvature and conservation laws

TL;DR

This paper demonstrates the existence of special orthonormal vector fields on negatively curved Riemannian manifolds, leading to conservation laws for certain PDEs like the Camassa-Holm and Sine-Gordon equations.

Contribution

It extends the theory of conservation laws on negatively curved manifolds to higher dimensions and constructs explicit orthonormal vector fields with geometric properties.

Findings

Existence of orthonormal vector fields tangent to geodesics and horocycles

The dual 1-form to the geodesic-tangent vector field is closed

Application to conservation laws for specific PDEs

Abstract

We show that any $n$-dimensional Riemannian manifold with constant negative sectional curvature admits local orthonormal vector fields such that one of them $v_1$ is tangent to geodesics and the other $n-1$ vector fields are tangent to horocycles. We prove that the $1$-form dual to $v_1$ is a closed form. We show how the closed form can be used to obtain conservation laws for PDEs whose generic solutions define metrics on open subsets with constant negative sectional curvature. These results extend to higher dimensions the $2$-dimensional case proved in the 1980s. We prove that there exist local coordinates on the manifold such that the coordinate curves are tangent to the orthonormal vector fields. We apply the theory to obtain conservation laws for the Camassa-Holm equation ($n=2$) and for the Intrinsic Generalized Sine-Gordon equation ($n\geq 2$).