On a class of K\"ahler manifolds whose geodesic flows are integrable

TL;DR

This paper investigates a special class of K"ahler manifolds with integrable geodesic flows, characterized by specific first integrals, leading to their classification as toric varieties under certain conditions.

Contribution

It introduces a new class of K"ahler manifolds with integrable geodesic flows defined by fibrewise hermitian forms and links them to toric varieties.

Findings

Existence of an n-dimensional commutative Lie algebra of automorphisms.

Geodesic flow integrability due to first integrals and automorphisms.

Compact manifolds in this class are toric varieties.

Abstract

We study $n$-dimensional K\"ahler 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.