1. CJM 2011 (vol 64 pp. 778)
|Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces|
We study the geometry of non-reductive $4$-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.
Keywords:non-reductive homogeneous spaces, pseudo-Riemannian metrics, Ricci solitons, Einstein-like metrics
Categories:53C21, 53C50, 53C25
2. CJM 2009 (vol 61 pp. 1201)
|Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups |
A Riemannian manifold $(M,\rho)$ is called Einstein if the metric $\rho$ satisfies the condition \linebreak$\Ric (\rho)=c\cdot \rho$ for some constant $c$. This paper is devoted to the investigation of $G$-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces $G/H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds $\SO(n)/\SO(l)$. Furthermore, we show that for any positive integer $p$ there exists a Stiefel manifold $\SO(n)/\SO(l)$ that admits at least $p$ $\SO(n)$-invariant Einstein metrics.
Keywords:Riemannian manifolds, homogeneous spaces, Einstein metrics, Stiefel manifolds
3. CJM 2005 (vol 57 pp. 1012)
|Deformations of $G_2$ and $\Spin(7)$ Structures |
We consider some deformations of $G_2$-structures on $7$-manifolds. We discover a canonical way to deform a $G_2$-structure by a vector field in which the associated metric gets ``twisted'' in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field $w$ whose solution would yield a manifold with holonomy $G_2$. Similarly we consider analogous constructions for $\Spin(7)$-structures on $8$-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the $\Spin(7)$ case.
Keywords:$G_2 \Spin(7)$, holonomy, metrics, cross product