Part 1. Liouville Manifolds Introduction Since Jacobi it is well known that the geodesic flows of ellipsoids are integrable in the sense of symplectic geometry (or in LiouviUe's sense). In fact they possess n (= the dimension of the ellipsoid) first integrals in involution that are quadratic forms on each cotangent space. Moreover those first integrals have a remarkable property they are simultaneously normaUzable on each fibre. If one uses the so- called elliptic coordinate system to describe the geodesic equation, then those first integrals are expressed in normalized form (and the variables are separated). After Jacobi's work, Liouville obtained a general form of integrable hamiltonian systems that are integrated in a similar way as the geodesic flows of ellipsoids, which we refer to as Liouville's system (see [L] pp. 700-708). Liouville's system (the case without potential) is expressed as follows: Let (a?i,..., xn, f 1 , . . . , £n) be a canonical coordinate system, and let bij(x{) be n2 functions in one variable such that det(6tj) ^ 0. Then the functions F{ defined by 3 satisfies {Fi,Fj} 0. (cf. Proposition 1.1.3.) Here, for example, F\ is the hamil- tonian, and Fi, ..., Fn are first integrals of the system. Later, Stackel characterized Liouville's system in terms of the properties of the first integrals. Namely, let Fi = H, F2, ..., Fn be a set of functions in involution on the cotangent bundle of n-dimensional manifold. Assume that every F{ is a quadratic form, and those quadratic forms are linearly independent and simultane- ously normaUzable on each cotangent space. Then one can find a coordinate system with which the hamiltonian system defined by the hamiltonian H is described as LiouviUe's system (cf. KUngenberg [Kll] and Proposition 1.1.3). The main purpose of this part is to investigate global properties of complete riemannian manifolds whose geodesic flows are regarded as LiouviUe's system (at least on a dense subset of the cotangent bundle). We caU the pair of such a rie- mannian manifold and the space of first integrals Liouville manifold. The precise definition is as foUows: Let M be an n-dimensional complete riemannian manifold, and let E be the associated energy function (the hamiltonian of the geodesic flow). Let T be an n-dimensional vector space of functions on the cotangent bundle T*M that are fibrewise homogeneous polynomials of degree two. We say that the pair (M, T) is a Liouville manifold if the foUowing four conditions are satisfied: (L.l) EeT (L.2) {F, H} = 0 for any F,H T\ (L.3) Fp (F T) are simultaneously normaUzable for any p £ M
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