Wallach Spaces Research Articles

On the Dynamics of a Three-dimensional Differential System Related to the Normalized Ricci Flow on Generalized Wallach Spaces

On the Dynamics of a Three-dimensional Differential System Related to the Normalized Ricci Flow on Generalized Wallach Spaces

New Einstein-Randers metrics on certain homogeneous manifolds arising from the generalized Wallach spaces

New Einstein-Randers metrics on certain homogeneous manifolds arising from the generalized Wallach spaces

On the stability of homogeneous Einstein manifolds II

For any G $G$ -invariant metric on a compact homogeneous space M = G / K $M=G/K$ , we give a formula for the Lichnerowicz Laplacian restricted to the space of all G $G$ -invariant symmetric 2-tensors in terms of the structural constants of G / K $G/K$ . As an application, we compute the G $G$ -invariant spectrum of the Lichnerowicz Laplacian for all the Einstein metrics on most generalized Wallach spaces and any flag manifold with b 2 ( M ) = 1 $b_2(M)=1$ . This allows to deduce the G $G$ -stability and critical point types of each of such Einstein metrics as a critical point of the scalar curvature functional.

Standard homogeneous (α1,α2)${\big(\alpha _1,\alpha _2\big)}$‐metrics and geodesic orbit property

AbstractIn this paper, we introduce the notion of standard homogeneous ‐metric, as a generalization for the normal homogeneity in Finsler geometry with relatively good computability. We explore the geodesic orbit (g.o. in short) property of this metric. Especially, when it is associated with a triple of compact Lie groups, we find new examples of g.o. Finsler spaces from H. Tamaru's classification list. Meanwhile, we prove that all standard g.o. ‐metrics on the Wallach spaces, , and , must be the normal homogeneous Riemannian metrics.

Spin(7) Metrics of Cohomogeneity One with Aloff–Wallach Spaces as Principal Orbits

In this article, we construct two continuous 1-parameter family of non-compact $$\mathrm {Spin}(7)$$ metrics with both chiralities, with the principal orbit an Aloff–Wallach space $$N_{k,l}$$ and the singular orbit $${{\mathbb {C}}}{{\mathbb {P}}}^2$$ . For a generic $$N_{k,l}$$ , metrics constructed are locally asymptotically conical (ALC). For $$N_{1,1}$$ , we construct two continuous 1-parameter families with geometric transition from asymptotically conical (AC) metrics to ALC metrics.

Equigeodesics on some classes of homogeneous spaces

We study homogeneous curves on some classes of reductive homogeneous spaces G/H which are geodesics with respect to any G-invariant metric on G/H. These curves are called equigeodesics. The spaces we consider are certain Stiefel manifolds VkRn, generalized Wallach spaces and spheres. We give a characterization for algebraic equigeodesics on V2Rn, V4R6, SO(6)/SO(3)⋅SO(2), W6=U(3)/U(1)3, W12=Sp(3)/Sp(1)3, S2n+1≅U(n+1)/U(n) and S4n+3≅Sp(n+1)/Sp(n).

Correction to: Classification of generalized Wallach spaces

Correction to: Classification of generalized Wallach spaces

On the moduli spaces of metrics with nonnegative sectional curvature

The Kreck–Stolz \(s\) invariant is used to distinguish connected components of the moduli space of positive scalar curvature metrics. We use a formula of Kreck and Stolz to calculate the \(s\) invariant for metrics on \(S^n\) bundles with nonnegative sectional curvature. We then apply it to show that the moduli spaces of metrics with nonnegative sectional curvature on certain 7-manifolds have infinitely many path components. These include certain positively curved Eschenburg and Aloff–Wallach spaces.

Invariant Ricci-flat metrics of cohomogeneity one with Wallach spaces as principal orbits

We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with respective principal orbits $G/K$ the Wallach spaces $SU(3)/T^2$, $Sp(3)/(Sp(1)Sp(1)Sp(1))$ and $F_4/\mathrm{spin}(8)$. Almost all the Ricci-flat metrics constructed have generic holonomy. The only exception is the complete $G_2$ metric discovered in [BS89][GPP90]. It lies in the interior of the 1-parameter family on $\bigwedge_-^2\mathbb{CP}^2$. All the Ricci-flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric on $G/K$.

О вырожденных особых точках динамических систем, имеющих отношение к нормализованному потоку Риччи на обобщенных пространствах Уоллаха

In this paper, we study degenerated singular points (equilibrium points) of a dynamical system obtained by reduction of the normalized Ricci flow on generalized Wallach spaces. It is known that every generalized Wallach space is characterized by a triple of positive numbers satisfying well-defined inequalities. Therefore, the corresponding system of differential equations also depends on three real parameters. In the works of N.A. Abiev, A. Arvanitoyeorgos, Yu.G. Nikonorov, and P. Siasos a new approach was developed for studying of singular points. This approach is based on the idea of building a surface of parameters providing the normalized Ricci flow degenerate singular points. At natural (geometric) values of parameters, we established that for the normalized Ricci flow the nilpotent case never occurs, and the linearly zero case can occur only at a unique special combination of parameters. As a consequence, any other degenerate singular point of the system may be only semi-hyperbolic. In this paper, we remove the previous restrictions and study an abstract dynamical system abstracted from geometric essence. It is proved that some results of the mentioned works also hold for arbitrary values of the real parameters.

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.

New non-naturally reductive Einstein metrics on exceptional simple Lie groups

In this article, we achieved several non-naturally reductive Einstein metrics on exceptional simple Lie groups, which are formed by the decomposition arising from general Wallach spaces. By using the decomposition corresponding to the two involutive automorphisms, we calculated the non-zero coefficients in the expression for the components of Ricci tensor with respect to the given metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gr\"{o}bner bases.

On the evolution of invariant Riemannian metrics on one class of generalized Wallach spaces under the influence of the normalized Ricci flow

The paper is devoted to the study of the evolution of invariant Riemannian metrics on the special class of generalized Wallach spaces corresponding to the case of a1 = a2 = a3 = 1/4. We prove that the normalized Ricci flow evolves all generic invariant Riemannian metrics into metrics with positive Ricci curvature.

Homogeneous Geodesics in Generalized Wallach Spaces

We classify generalized Wallach spaces which are g.o. spaces. We also investigate homogeneous geodesics in generalized Wallach spaces for any given invariant Riemannian metric and we give some examples.

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.

Sasakian quiver gauge theory on the Aloff–Wallach space X1,1

We consider the SU(3)-equivariant dimensional reduction of gauge theories on spaces of the form Md×X1,1 with d-dimensional Riemannian manifold Md and the Aloff–Wallach space X1,1=SU(3)/U(1) endowed with its Sasaki–Einstein structure. The condition of SU(3)-equivariance of vector bundles, which has already occurred in the studies of Spin(7)-instantons on cones over Aloff–Wallach spaces, is interpreted in terms of quiver diagrams, and we construct the corresponding quiver bundles, using (parts of) the weight diagram of SU(3). We consider three examples thereof explicitly and then compare the results with the quiver gauge theory on Q3=SU(3)/(U(1)×U(1)), the leaf space underlying the Sasaki–Einstein manifold X1,1. Moreover, we study instanton solutions on the metric cone C(X1,1) by evaluating the Hermitian Yang–Mills equation. We briefly discuss some features of the moduli space thereof, following the main ideas of a treatment of Hermitian Yang–Mills instantons on cones over generic Sasaki–Einstein manifolds in the literature.

Two-step Homogeneous Geodesics in Homogeneous Spaces

We study geodesics of the form $\gamma(t)=\pi(\exp(tX)\exp(tY))$, $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $\pi:G\rightarrow G/K$ is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of $G$ (i.e. $\gamma(t)=\pi(\exp (tX))$, $X\in \fr{g}$). We obtain sufficient conditions on a homogeneous space implying the existence of such geodesics for $X,Y\in \fr{m}=T_o(G/K)$. We use these conditions to obtain examples of Riemannian homogeneous spaces $G/K$ so that all geodesics of $G/K$ are of the above form. These include total spaces of homogeneous Riemannian submersions endowed with one parameter families of fiber bundle metrics, Lie groups endowed with special one parameter families of left-invariant metrics, generalised Wallach spaces, generalized flag manifolds, and $k$-symmetric spaces with $k$-even, equipped with certain one-parameter families of invariant metrics.

The evolution of positively curved invariant Riemannian metrics on the Wallach spaces under the Ricci flow

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, and $F_4/Spin(8)$. We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature. Moreover, we prove that for the spaces $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$ and $F_4/Spin(8)$, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive Ricci curvature into metrics with mixed Ricci curvature. We also get similar results for some more general homogeneous spaces.

Classification of generalized Wallach spaces

In this paper, we present the classification of generalized Wallach spaces and discuss some related problems.

Homogeneous Einstein–Randers metrics on Aloff–Wallach spaces

Aloff–Wallach spaces are important in the study of positively curved homogeneous Riemannian manifolds. In this paper, we find some homogeneous Einstein–Randers metrics on such spaces.