On Hermite-Liouville manifolds

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In this paper we study a certain class of Hermitian n-manifolds whose geodesic flows admit n first integrals of certain kind. It is a generalization of Kähler-Liouville manifold in [3] and called Hermite-Liouville manifold. We completely determine the local structure of Hermite-Liouville manifolds “of type (A)”, and construct global examples over the complex projective space.