Curvature-homogeneous indefinite Einstein metrics in dimension four: the diagonalizable case

Classification of homogeneous Einstein metrics on pseudo-hyperbolic spaces

We classify the effective and transitive actions of a Lie group $G$ on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that $G$ is a closed, connected Lie subgroup of $SO_0(n-r,r+1)$, the connected component of the indefinite special orthogonal group. Assuming additionally that $G$ acts completely reducible on $\mathbb R^{n+1}$, we also obtain that any $G$-homogeneous Einstein pseudo-Riemannian metric on a real, complex or quaternionic pseudo-hyperbolic space, or on a para-complex or para-quaternionic projective space is homothetic t…

Towards a Classification of Homogeneous Tube Domains in C^4

We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains (that do not have bounded realisations). These domains lie to either side of a pair of Levi-indefinite hypersurface. Using the geometry of these two hypersurfaces, we find the automorphism groups of the domains.

Diagram involutions and homogeneous Ricci-flat metrics

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups ${\rm SL}(n)$, ${\rm SO}(p,q)$, ${\rm Sp}(n,\mathbb R)$. Most of these metrics are shown not to be flat.

Ricci flow and curvature on the variety of flags on the two dimensional projective space over the complexes, quaternions and the octonions

For homogeneous metrics on the spaces of the title it is shown that the Ricci flow can move a metric of stricly positive sectional curvature to one with some negative sectional curvature and one of positive definite Ricci tensor to one with indefinite signature.

Metric geometries over the split quaternions

We give an overview of some recent results in hypersymplectic and para-quaternionic Kahler geometry, and introduce the notion of split three-Sasakian manifold. In particular, we discuss the twistor spaces and Swann bundles of para-quaternionic Kahler manifolds. These are used to classify examples with a fully homogeneous action of a semi-simple Lie group, and to construct distinct para-quaternionic Kahler metrics from indefinite real analytic conformal manifolds. We also indicate how the theory of toric varieties gives rise to constructions of hypersymplectic manifolds.

旗流形的通用嵌入与刚性现象

Universal embeddings of flag manifolds and rigidity phenomena

本研究证明旗流形可全纯等距嵌入不可约经典旗流形，借此建立齐性凯勒流形的新刚性现象，推广Loi-Mossa刚性定理并统一已有相关结果。

紧 Weyl 平行流形

Compact Weyl-parallel manifolds

本文概述与 Ivo Terek 的合作证明，说明非局部齐次的紧秩一 ECS 流形在取二重等距覆盖后必为圆周上丛的全空间。

具有平行Weyl曲率的紧平面波

Compact plane waves with parallel Weyl curvature

本文阐述与Andrzej Derdzinski合作的本质共形对称（ECS）流形研究成果，回顾其基本性质，讨论紧例子构造，证明紧秩一ECS流形的拓扑结构定理，并提及相关分类结果。

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。