First integrals of affine connections and Hamiltonian systems of hydrodynamic type
Differential Geometry
2016-08-29 v2 High Energy Physics - Theory
Exactly Solvable and Integrable Systems
Abstract
We find necessary and sufficient conditions for a local geodesic flow of an affine connection on a surface to admit a linear first integral. The conditions are expressed in terms of two scalar invariants of differential orders 3 and 4 in the connection. We use this result to find explicit obstructions to the existence of a Hamiltonian formulation of Dubrovin--Novikov type for a given one--dimensional system of hydrodynamic type. We give several examples including Zoll connections, and Hamiltonian systems arising from two--dimensional Frobenius manifolds.
Keywords
Cite
@article{arxiv.1510.01906,
title = {First integrals of affine connections and Hamiltonian systems of hydrodynamic type},
author = {Felipe Contatto and Maciej Dunajski},
journal= {arXiv preprint arXiv:1510.01906},
year = {2016}
}
Comments
11 pages. Final version. The analysis of sufficient conditions expanded