(TE)-structures over the irreducible 2-dimensional globally nilpotent F-manifold germ

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。

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…

An application of Cartan's equivalence method to Hirschowitz's conjecture on the formal principle

A conjecture of Hirschowitz's predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle, i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartan's equivalence method to a suitable differential system on the universal family of the Douady space of the complex manifold, we prove that this conjecture is true if $A$ is a Fano manifold, or if the global sections of $W$ separate points of $A$. Our method shows more generally that f…

Trivializable and quaternionic subriemannian structure on $\mathbb{S}^7$ and subelliptic heat kernel

On the seven dimensional Euclidean sphere $\mathbb{S}^7$ we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution $\mathcal{H}_T$ is induced by a Clifford module structure of $\mathbb{R}^8$. More precisely, $\mathcal{H}_T$ is rank $4$, bracket generating of step two and generated by globally defined vector fields. The distribution $\mathcal{H}_{Q}$ of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fi…

量子信息的无穷小结构

The Infinitesimal Structure of Quantum Information

该研究为量子态空间建立严格统一几何框架，通过嵌入高阶对偶数代数，将量子态表示为非约化概型理论点，使控制刘维尔 - 冯·诺依曼动力学的换位子映射到代数流上，确立幂零对偶代数为高维量子运动学几何图景。