Filiform nilsolitons of dimension 8

Totally Geodesic Subalgebras of Filiform Nilpotent Lie algebras

We determine the maximal dimension of totally geodesic subalgebras of N-graded filiform Lie algebras, and we show that these bounds are attained.

The space of solsolitons in low dimensions

Up to now, the only known examples of homogeneous nontrivial Ricci soliton metrics are the so called solsolitons, i.e. certain left invariant metrics on simple connected solvable Lie groups. In this paper, we describe the moduli space of solsolitons of dimension less or equal than 6, up to isomorphism and scaling. We start with the already known classification of nilsolitons and, following the characterization given by Lauret, we describe the subspace of solsolitons associated to a given nilsoliton, as the quotient of a Grassmanian by a finite group.

Ricci soliton solvmanifolds

All known examples of nontrivial homogeneous Ricci solitons are left-invariant metrics on simply connected solvable Lie groups whose Ricci operator is a multiple of the identity modulo derivations (called solsolitons, and nilsolitons in the nilpotent case). The tools from geometric invariant theory used to study Einstein solvmanifolds, turned out to be useful in the study of solsolitons as well. We prove that, up to isometry, any solsoliton can be obtained via a very simple construction from a nilsoliton together with any abelian Lie algebra of symmetric derivations of its metric Lie algebra. …

Solvable extensions of negative Ricci curvature of filiform Lie groups

We give necessary and sufficient conditions of the existence of a left-invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra $\mathfrak{g}$ is a filiform Lie algebra $\mathfrak{n}$. It turns out that such a metric always exists, except for in the two cases, when $\mathfrak{n}$ is one of the algebras of rank two, $L_n$ or $Q_n$, and $\mathfrak{g}$ is a one-dimensional extension of $\mathfrak{n}$, in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation.

Graded filiform Lie algebras and symplectic nilmanifolds

We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras - nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e_1, ..., e_n, [e_i,e_j]=c_{ij}e_{i{+}j} (N-graded Lie algebras). In particular we describe the spaces of symplectic cohomology classes for all even-dimensional algebras of the list. It is proved that a symplectic filiform Lie algebra is a filtered deformation of some N-graded symplectic filiform Lie algebra. But this condition is no…

李群的一维和二维交换子群的黎曼几何

Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two

本文研究了具有不同维度交换子群的李群上的黎曼几何性质，包括里奇流形的条件和特征。

伪黎曼可解流形上的爱因斯坦度量和Killing旋量

Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds

本文通过nilsoliton构造了一类任意维数和符号的伪Iwasawa可解流形，证明了所有允许Killing旋量的伪Iwasawa可解流形都属于该族，并指出当度量爱因斯坦时唯一可能是双曲半空间。

可解李代数上的新特殊爱因斯坦伪黎曼度量

New Special Einstein Pseudo-Riemannian Metrics on Solvable Lie Algebras

本研究提出可解李代数上爱因斯坦伪凯勒与仿凯勒度量的构造方法，完成低维(Nil4)型幂零孤立子的秩一伪伊瓦萨瓦扩张分类，证明该度量在广义海森堡李代数对应扩张中存在，并给出后续研究思路。