Some Examples of the Hermite-Liouville Structure on the Classical Hopf Surface

Masayuki Igarashi


Abstract. It is rewarding to investigate compact riemannian manifolds whose geodesic flows are completely integrable. A Hermite-Liouville structure on a complex surface is defined as a couplet $( g, L )$ of a hermitian metric $g$ and a 2-dimensional real vector space $L$ of first integrals on the cotangent bundle of its geodesic flow such that (1) $L$ contains the energy function (2) all the first integrals contained in $L$ are fiberwise homogenous polynomials of degree 2 and are hermitian. The author constructs some examples of the Hermite-Liouville structure on the classical Hopf surface. The geodesic flows of these examples are completely integrable.

AMSclassification. 53, 22

Keywords. Riemannian manifold, geodesic, geodesic flow, complete integrability