标题： 斯塔克尔和埃森哈特提升，哈恩杰斯几何与引力 显示英文标题

标题： Stäckel and Eisenhart lifts, Haantjes geometry and Gravitation

摘要： 我们引入了Stäckel提升的概念，作为构建大量可积哈密顿系统的新几何设置。 Stäckel提升扩展了黎曼型和洛伦兹型经典Eisenhart提升背后的几何框架，并与Haantjes几何密切相关。 特别是，我们证明通过Stäckel提升构造的哈密顿系统自然地具有辛-Haantjes结构。 我们还表明，显式依赖动量的提升矩阵会产生可解释为引力波或哈密顿和Finsler几何中依赖动量的度量的系统，可能在修改引力理论中有应用。 显示英文摘要

摘要： We introduce the notion of St\"ackel lift as a novel geometric setting for the construction of large classes of integrable Hamiltonian systems. The St\"ackel lift extends the geometric framework underlying both the Riemannian and the Lorentzian-type classical Eisenhart lifts and is intimately related with Haantjes geometry. In particular, we establish that Hamiltonian systems constructed through St\"ackel lifts are naturally endowed with a symplectic-Haantjes structure. We also show that explicitly momentum-dependent lifting matrices produce systems interpretable as gravitational waves, or momentum-dependent metrics of Hamilton and Finsler geometries, with potential applications in modified gravity theories.