An integrability condition for fields of nilpotent endomorphisms

On integrability of transverse Lie-Poisson structures at nilpotent elements

We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform construction of these integrable systems for an infinite family of distinguished nilpotent elements of semisimple type.

The algebra of parallel endomorphisms of a germ of pseudo-Riemannian metric

On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e. sections of End(TM) may be parallel for g. They form an associative algebra A, which is also the commutant of the holonomy group of g. As any associative algebra, A is the sum of its radical and of a semi-simple algebra S. We show in arXiv:1402.6642 that S may be of eight different types, including the generic type S=R.Id, and the Kähler and hyperkähler types where S is respectively isomorphic to the complex field C or to the quaternions H. We show here that for any self adjoint nilpotent element N of the commutant of s…

亚阿贝尔幂零群的次黎曼测地流之辛约化

Sympletic reduction of the sub-Riemannian geodesic flow for metabelian nilpotent groups

本文研究导子群为阿贝尔群的幂零李群上的次黎曼正规哈密顿流，通过辛约化建立正规轨迹与多项式哈密顿量的对应，并给出可积性判据及全局长度极小化条件。

平行张量的Nijenhuis几何

Nijenhuis geometry of parallel tensors

本文研究了平行张量的可积性，通过代数常数性和 Nijenhuis 型张量的消失来特征化不同类型的张量。