Classification of invariant Einstein metrics on certain compact homogeneous spaces

In this paper, we study invariant Einstein metrics on certain compact homogeneous spaces with three isotropy summands. We show that, if G/K is a compact isotropy irreducible space with G and K simple,then except for some very special cases, the coset space G × G/?(K) carries at least two invariant Einstein metrics. Furthermore, in the case that G_1, G_2 and K are simple Lie groups, with K■G_1, K■G_2, and G_1≠G_2, such that G_1/K and G_2/K are compact isotropy irreducible spaces, we give a complete classification of invariant Einstein metrics on the coset space G_1 × G_2/?(K).     

Einstein metrics on even-dimensional homogeneous spaces admitting a homogeneous Riemannian metric of positive sectional curvature

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 126–131, May–June, 1991.     

Singularity of homogeneous Einstein metrics

We examine homogeneous cosmological models with arbitrary (uniform) motion of matter. We have shown the presence of an oscillatory mode of the BLK type when moving toward a cosmological singularity in models of II–IV and VI–IX Bianchi types. We have formulated constraints on the velocities under which the oscillatory mode degenerates to Kästner asymptotics.     

New series of Einstein homogeneous metrics

Some new examples of homogeneous Einstein metrics are constructed using the stability of nondegenerate critical points of smooth functions on some manifolds.     

Invariant Einstein metrics on generalized Wallach spaces

Invariant Einstein metrics on generalized Wallach spaces have been classified except SO(k + l+ m)/SO(k) × SO(l) × SO(m). In this paper, we first give a survey on the study of invariant Einstein metrics on generalized Wallach spaces, and prove that there are infinitely many spaces of the type SO(k + l + m)/SO(k)× SO(l) × SO(m) admitting exactly two, three, or four invariant Einstein metrics up to a homothety.     

Spinor structures on homogeneous spaces

An algebraic criterion for the existence of spinor structures on homogeneous spaces used in multidimensional models is developed. A method of explicit construction of spinor structures is proposed, and its effectiveness is demonstrated in examples.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 13–23, July, 1992.     

Geodesic orbit metrics on homogeneous spaces constructed by strongly isotropy irreducible spaces

In this paper, we focus on homogeneous spaces which are constructed from two strongly isotropy irreducible spaces, and prove that any geodesic orbit metric on these spaces is naturally reductive.     

Half-flat structures on decomposable lie groups

Half-at SU(3)-structures are the natural initial values for Hitchin’s evolution equations whose solutions define parallel G₂-structures. Together with the results of [SH], the results of this article completely solve the existence problem of left-invariant half-at SU(3)-structures on decomposable Lie groups. The proof is supported by the calculation of the Lie algebra cohomology for all indecomposable five-dimensional Lie algebras, which refines and clarifies the existing classification of five-dimensional Lie algebras.     

Half-flat structures on indecomposable Lie groups

This article can be viewed as a continuation of the articles [SH] and [FS] in which the decomposable Lie algebras admitting half-flat SU(3)-structures are classified. The new main result is the classification of the indecomposable six-dimensional Lie algebras with five-dimensional nilradicals which admit a half-flat SU(3)-structure. As an important step of the proof, a considerable refinement of the classification of six-dimensional Lie algebras with five-dimensional non-Abelian nilradicals is established. Additionally, it is proved that all non-solvable six-dimensional Lie algebras admit half-flat SU(3)-structures.     

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres ${S^n}$ . We prove that for any connected (almost effective) transitive on $S^n$ compact Lie group $G$ , the family of $G$ -invariant Riemannian metrics on $S^n$ contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and $n\ge 5$ . Any such family (that exists only for $n=2k+1$ ) contains a metric $g_\mathrm{can}$ of constant sectional curvature $1$ on $S^n$ . We also prove that $(S^{2k+1}, g_\mathrm{can})$ is Clifford–Wolf homogeneous, and therefore generalized normal homogeneous, with respect to $G$ (except the groups $G={ SU}(k+1)$ with odd $k+1$ ). The space of unit Killing vector fields on $(S^{2k+1}, g_\mathrm{can})$ from Lie algebra $\mathfrak g $ of Lie group $G$ is described as some symmetric space (except the case $G=U(k+1)$ when one obtains the union of all complex Grassmannians in $\mathbb{C }^{k+1}$ ).     

TheL 2 structure of moduli spaces of Einstein metrics on 4-manifolds

Partially supported by NSF Grants DMS 87-01137 and 89-01303     

Einstein metrics and the number of smooth structures on a four-manifold

We prove that for every natural number k there are simply connected topological four-manifolds which have at least k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic to each other after connected sum with only one copy of the complex projective plane. We prove that manifolds with these properties cover a large geographical area.