Invariant Einstein Metrics Research Articles

Non Naturally Reductive Einstein Metrics on the Orthogonal Group via Real Flag Manifolds

We obtain new invariant Einstein metrics on the compact Lie groups SO(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(n)$$\\end{document} which are not naturally reductive. This is achieved by using the real flag manifolds SO(k1+⋯+kp)/SO(k1)×⋯×SO(kp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(k_1+\\cdots +k_p)/{{\\,\ extrm{SO}\\,}}(k_1)\ imes \\cdots \ imes {{\\,\ extrm{SO}\\,}}(k_p)$$\\end{document} and by imposing certain symmetry assumptions in the set of all left-invariant metrics on SO(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(n)$$\\end{document}.

LOCAL PARAMETRIZATION OF AN ALGEBRAIC VARIETY NEAR SOME ITS SINGULARITIES

In theoretical physics, when studying invariant Einstein metrics, there was a need to study an algebraic variety, which is described by an equation of the twelfth degree in three variables. 4 singular points and 2 curves of singular points of the variety were found early. Using algorithms of power geometry and computer algebra programs, local parametrizations of a manifold near 2 its singular points of the third order and near one curve of singular points are obtained.

Invariant Einstein metrics on real flag manifolds with two or three isotropy summands

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible subrepresentations. In this situation, one can have equivalent submodules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds.

English

We prove that there are at least two new non-naturally reductive Ad(Sp(l) × Sp(k) × Sp(k) × Sp(k)) invariant Einstein metrics on Sp(l + 3k) (k 4 admits at least $$2[{1 \over 4}(n - 1)]$$ non-naturally reductive Ad(Sp(l) × Sp(k) × Sp(k) × Sp(k)) invariant Einstein metrics.

Ancient solutions of the homogeneous Ricci flow on flag manifolds

For any flag manifold M=G/K of a compact simple Lie group G we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions emerge from an invariant Einstein metric on M, and by [13] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold M=G/K with second Betti number b2(M) = 1, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose α-limit set consists of fixed points at infinity of MG. Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system’s phase space.

Homogeneous Einstein metrics on non-Kähler C-spaces

We study homogeneous Einstein metrics on indecomposable non-Kähler C-spaces, i.e. even-dimensional torus bundles M=G∕H with rankG>rankH over flag manifolds F=G∕K of a compact simple Lie group G. Based on the theory of painted Dynkin diagrams we present the classification of such spaces. Next we focus on the family Mℓ,m,n≔SU(ℓ+m+n)∕SU(ℓ)×SU(m)×SU(n),ℓ,m,n∈Z+and examine several of its geometric properties. We show that invariant metrics on Mℓ,m,n are not diagonal and beyond certain exceptions their parametrization depends on six real parameters. By using such an invariant Riemannian metric, we compute the diagonal and the non-diagonal part of the Ricci tensor and present explicitly the algebraic system of the homogeneous Einstein equation. For general positive integers ℓ,m,n, by applying mapping degree theory we provide the existence of at least one SU(ℓ+m+n)-invariant Einstein metric on Mℓ,m,n. For ℓ=m we show the existence of two SU(2m+n)-invariant Einstein metrics on Mm,m,n, and for ℓ=m=n we obtain four SU(3n)-invariant Einstein metrics on Mn,n,n. We also examine the isometry problem for these metrics, while for a plethora of cases induced by fixed ℓ,m,n, we provide the numerical form of all non-isometric invariant Einstein metrics.

On Einstein Lorentzian nilpotent Lie groups

In this paper, we study Lorentzian left invariant Einstein metrics on nilpotent Lie groups. We show that if the center of such Lie groups is degenerate then they are Ricci-flat and their Lie algebras can be obtained by the double extension process from an abelian Euclidean Lie algebra. We show that all nilpotent Lie groups up to dimension 5 endowed with a Lorentzian Einstein left invariant metric have degenerate center and we use this fact to give a complete classification of these metrics. We show that if g is the Lie algebra of a nilpotent Lie group endowed with a Lorentzian left invariant Einstein metric with non zero scalar curvature then the center Z(g) of g is nondegenerate Euclidean, the derived ideal [g,g] is nondegenerate Lorentzian and Z(g)⊂[g,g]. We give the first examples of Ricci-flat Lorentzian nilpotent Lie algebra with nondegenerate center.

Homogeneous Einstein metrics on Stiefel manifolds associated to flag manifolds with two isotropy summands

We study invariant Einstein metrics on the Stiefel manifold VkRn≅SO(n)/SO(n−k) of all orthonormal k-frames in Rn. The isotropy representation of this homogeneous space contains equivalent summands, so a complete description of G-invariant metrics is not easy. In this paper we view the manifold V2pRn as total space over a classical generalized flag manifold with two isotropy summands and prove for 2≤p≤25n−1 it admits at least four invariant Einstein metrics determined by Ad(U(p)×SO(n−2p))-invariant scalar products. Two of the metrics are Jensen's metrics and the other two are new Einstein metrics. The Einstein equation reduces to a parametric system of polynomial equations, which we study by combining Gröbner bases and geometrical arguments.

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 $G\times~G/\Delta(K)$ carries at least two invariant Einstein metrics. Furthermore, in the case that $G_{1},~G_{2}$ and $K$ are simple Lie groups, with $K\subset~G_1,~K\subset~G_2$, and $G_{1}\neq~G_{2}$, 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 $G_{1}\times~G_{2}/\Delta(K)$.

New homogeneous Einstein metrics on quaternionic Stiefel manifolds

Abstract We consider invariant Einstein metrics on the quaternionic Stiefel manifold V p ℍ n of all orthonormal p-frames in ℍ n . This manifold is diffeomorphic to the homogeneous space Sp(n)/Sp(n − p) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on V p ℍ n ≅ Sp(n)/Sp(n − p), where n = k 1 + k 2 + k 3 and p = n − k 3. We view V p ℍ n as a total space over the generalized Wallach space Sp(n)/(Sp(k 1)×Sp(k 2)×Sp(k 3)) and over the generalized flag manifold Sp(n)/(U(p)×Sp(n − p)).

New non-naturally reductive Einstein metrics on SO(n)

We obtain new invariant Einstein metrics on the compact Lie groups [Formula: see text] ([Formula: see text]) which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on [Formula: see text] and by computing the Ricci tensor for such metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.

Invariant Einstein metrics on certain compact semisimple Lie groups

In this paper, we study left invariant Einstein metrics on compact semisimple Lie groups. A new method to construct holonomy irreducible non-naturally reductive Einstein metrics on certain compact semisimple (non-simple) Lie groups is presented. In particular, we show that if G is a classical compact simple Lie group and H is a closed subgroup such that G/H is a standard homogeneous Einstein manifold, then there exist holonomy irreducible non-naturally reductive Einstein metrics on H×G, except for some very special cases. A further interesting result of this paper is that for any compact simple Lie group G, there always exist holonomy irreducible non-naturally reductive Einstein metrics on the compact semisimple Lie groups Gn, for any n≥4.

Einstein metrics and Einstein–Randers metrics on a class of homogeneous manifolds

In this paper, we give [Formula: see text]-invariant Einstein metrics on a class of homogeneous manifolds [Formula: see text], and then prove that every homogeneous manifold [Formula: see text] admits at least three families of [Formula: see text]-invariant non-Riemannian Einstein–Randers metrics.

Invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$

It is well known that the Einstein equation on a Riemannian flag manifold $(G/K,g)$ reduces to an algebraic system if $g$ is a $G$-invariant metric. In this paper we obtain explicitly new invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$; and we compute the Einstein system for generalized flag manifolds of type $Sp(n)$. We also consider the isometric problem for these Einstein metrics.

Invariant Einstein metrics on generalized Wallach spaces

Invariant Einstein metrics on generalized Wallach spaces have been classified except $SO(k+l+m)/SO(k)\times SO(l)\times SO(m)$. In this paper, we 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 $SO(k+l+m)/SO(k)\times SO(l)\times SO(m)$ admitting exactly two, three, or four Einstein invariant metrics up to a homothety.

A real variety with boundary and its global parameterization

An algebraic variety in R3 is studied that plays an important role in the investigation of the normalized Ricci flow on generalized Wallach spaces related to invariant Einstein metrics. A procedure for obtaining a global parametric representation of this variety is described, which is based on the use of the intersection of this variety with the discriminant set of an auxiliary cubic polynomial as the axis of parameterization. For this purpose, elimination theory and computer algebra are used. Three different parameterization of the variety are obtained; each of them is valid for certain noncritical values of one of the parameters.

Homogeneous Einstein Metrics on Certain Generalized Flag Manifolds with Six Isotropy Summands

For a generalized flag manifold M = G/K of a compact connected simple Lie group G whose isotropy representation decomposes into more than five isotropy summands, there are only a few results about the homogeneous Einstein metrics on M. Finding the invariant Einstein metrics on generalized flag manifolds, there are two difficulties. One is computing the non-zero structure constants, the other is computing the Grobner basis of the system of Einstein equations. In this paper, we give a method (Theorem A) which can be used to calculate structure constants of generalized flag manifolds with any number of isotropy summands. In this direction we present invariant Einstein metrics on some flag manifolds of exceptional groups with six isotropy summands.

New homogeneous Einstein metrics on Stiefel manifolds

We consider invariant Einstein metrics on the Stiefel manifold VqRn of all orthonormal q-frames in Rn. This manifold is diffeomorphic to the homogeneous space SO(n)/SO(n−q) and its isotropy representation contains equivalent summands. We prove, by assuming additional symmetries, that V4Rn(n⩾6) admits at least four SO(n)-invariant Einstein metrics, two of which are Jensen's metrics and the other two are new metrics. Moreover, we prove that V5R7 admits at least six invariant Einstein metrics, two of which are Jensen's metrics and the other four are new metrics.

The classification of homogeneous Einstein metrics on flag manifolds with [formula omitted

Consider a compact connected simple Lie group G. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds G/H with second Betti number b2(G/H)=1. There are 8 infinite families G/H corresponding to a classical simple Lie group G and 25 exceptional flag manifolds, which all have some common geometric features; for example they admit a unique invariant complex structure which gives rise to unique invariant Kähler–Einstein metric. The most typical examples are the compact isotropy irreducible Hermitian symmetric spaces for which the Killing form is the unique homogeneous Einstein metric (which is Kähler). For non-isotropy irreducible spaces the classification of homogeneous Einstein metrics has been recently completed for 24 of the 26 cases. In this paper we construct the Einstein equation for the two unexamined cases, namely the flag manifolds E8/U(1)×SU(4)×SU(5) and E8/U(1)×SU(2)×SU(3)×SU(5). In order to determine explicitly the Ricci tensors of an E8-invariant metric we use a method based on the Riemannian submersions. For both spaces we classify all homogeneous Einstein metrics and thus we conclude that any flag manifold G/H with b2(M)=1 admits a finite number of non-isometric non-Kähler invariant Einstein metrics. The precise number of these metrics is given in Table 1.

HOMOGENEOUS EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH FIVE ISOTROPY SUMMANDS

We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad (K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E6/(SU(4) × SU(2) × U(1) × U(1)) and E7/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO (2ℓ + 1)/( U (1) × U (p) × SO (2(ℓ - p - 1) + 1)) and SO (2ℓ)/( U (1) × U (p) × SO (2(ℓ - p - 1))) we prove existence of at least two non-Kähler–Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.