We show that any n-dimensional Riemannian manifold with constant negative sectional curvature admits local orthonormal vector fields such that one of them v₁ is tangent to geodesics and the other n-1 vector fields are tangent to horocycles. We prove that the 1-form dual to v₁ 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 2).
Tenenblat et al. (Tue,) studied this question.