Two classes of Riemannian manifolds whose geodesic flows are integrable

Abstract

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.

On Liouville integrability of $h$-projectively equivalent Kähler metrics

“…is a Kähler-Liouville manifold if the following conditions are satisfied (see [1,Part 2,Introduction]). …”

“…Let G be the transformation group of M generated by g = h + Jh. The group G is isomorphic to (C × ) n , and with this action M becomes a toric variety (see [1], Section 4). Since the associated partially ordered set consists of one element, we have the following (cf.…”

“…As stated in [3], some parts of [1] are incorrect unless an additional condition called properness is assumed. Note however that in the case where the associated partially ordered set consists of one element, the properness condition is automatically satisfied (see [3] for more details).…”

“…It was proven, among other things, that under a nondegeneracy condition such Riemannian manifolds have integrable geodesic flows. More specifically, it was shown that these manifolds are Liouville manifolds in the sense of [1]. Also, from a single pair of projectively equivalent metrics, a hierarchy of such pairs was constructed; see e.g.…”

“…Equation (1.1) implies that any holomorphically planar curve with respect to one of the metrics is holomorphically planar also with respect to the other metric and vice versa; see for example [10,Introduction,Example (b)], [4,7], for more details, as well as [11,12] for related work. It was shown in [10] that under a nondegeneracy condition Kähler manifolds admitting pairs of h-projective equivalent metrics are Kähler-Liouville manifolds in the sense of [1]. As in the real case, there exists a hierarchy of such pairs of metrics.…”