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

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*Masayuki Igarashi*

**E-mail:** igarashi@it.osha.sut.ac.jp
**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