Abstract. Under a nondegeneracy condition we classify the compact connected Kähler manifolds admitting pairs of h-projectively equivalent metrics. Any such manifold is biholomorphically equivalent to CP n and has integrable geodesic flow.
We study n-dimensional Kähler manifolds whose geodesic flows possess n first integrals in involution that are fibrewise hermitian forms and simultaneously normalizable. Under some mild assumption, one can associate with such a manifold an n-dimensional commutative Lie algebra of infinitesimal automorphisms. This, combined with the given n first integrals, makes the geodesic flow integrable. If the manifold is compact, then it becomes a toric variety.
We present a family of riemannian metrics on two-sphere having the property that the geodesic flows admit first integrals which are fiberwise homogeneous polynomials of degree greater than 2. They also have the property that all geodesics are closed. (2000): 53C22, 58F05
Mathematics Subject Classification
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